Glass 




Book_y7 



I8r33a^ 



ELEMENTS 

OF 

PLANE AND SPHERICAL 

TRIGONOMETRY, 

WITH 
ITS APPLICATIONS TO THE PRINCIPLES 

OF 

NAVIGATION AND NAUTICAL ASTRONOMY; 

WITH THE 

LOGARITHMIC AND TRIGONOMETRICAL TABLES. 
BY J. R. YOUNG, 

ADTHOR OF "THE ELEMENTS OF ANALYTICAL GEOMETRY," "ELEMENTS 
OF THE DIFFERENTIAL AND INTEGRAL CALCULUS," &C. 

TO WHICH ARE ADDED 
SOME ORIGINAL RESEARCHES IN 

SPHERICAL GEOMETRY ; 
BY T. S. DA VIES, F.R.S.E., F.R.A.S., &c. 



REVISED AND CORRECTED BY 

J. D. WILLIAMS, 

AUTHOR OF " KEY TO HDTTON'S MATHEMATICS," &C. 



4 



CAREY, LEA, & BLANCHARD. 



,0 



1833. 



sll 



Entered according to the act of congi-ess, in the year 1833, by Carey, 
Lea, & Blanchard, in the clerk's office of the district court of the eastern 
district of Pennsylvania. 



/c/'^ff 






. \ 



D 



ADVERTISEMENT 

TO 

THE AMERICAN EDITION. 



The extensive circulation and rapid sale which attended 
the republication, in this country, of Mr. Young's Treatise 
on Algebra, revised and corrected by Mr. Ward, the Ele- 
ments of Geometry, by M. Floy, Jr., Analytical Geometry, 
by the Editor of the present Treatise, and also the Ele- 
ments of the Differential and Integral Calculus, by Michael 
O'Shannessy, all of New- York, — together with the increas- 
ing demand for all his other Treatises upon the elements of 
abstract science, have induced me to attempt, by the publi- 
cation of the present work, to place, as it were, the key-stone 
of the arch of Elementary Mathematics. 

No further merit is claimed, in this edition, than that of a 
careful correction of all the errors occurring in the original 
and republished typography ; as also of having altered the 
first part of the valuable Trigonometrical Tables connected 
with the work, so as to correspond with the improved plan, 
which was not adopted by the author, as will be seen by his 
preface, until after some few sheets were stereotyped, and 
consequently past all recall. 

A consideration of the rapidly advancing state of ana- 
lytical science amongst us, must be soul-reviving to every 
lover of his country: — And tnat her sons may continue daily 
to increase in a scientific knowledge of the mysterious prin- 
ciples hidden so long in the vast book of nature, is the fer- 
vent aspiration of one of Science's most humble votaries. 

JOHN D. WILLIAMS. 
New- York, 1833. 



PREFACE. 



It is the design of this treatise to estabhsh the theory of Plane and 
Spherical Trigonometry analytically, and to present that theory, together 
with some of its most interesting and valuable applications, in a form 
fitted for elementary instruction. 

Of late years several analytical works on Trigonometry have been pub- 
lished in thfs country; but, as they are confined almost entirely to the 
theory of the subject, it may be questioned whether, to many young 
students, they prove much else than so many collections of mere alge- 
braical exercises. Yet a book upon so practical a subject as Trigonometry, 
ought undoubtedly to be something niore than this, and ought not to be 
considered as complete when the various calculations which the science 
involves, and which its name imphes, are wholly omitted. 

The symbolical expression of a practical rule, in algebraic language, will 
often, to the young student, but indistinctly point out the numerical opera- 
tion. Those much occupied in mathematical instruction, know full well 
that a learner may readily yield his assent to every stop of an algebraic 
process, be fully satisfied as to the truth of the result to which it leads, may 
even clearly see a valuable truth involved in it, and may yet be very far 
from perceiving how to turn it to account in any case of actual calculation. 
Indeed, algebraical formulas, transform them as we will, cannot always be 
made to indicate the best mode of arithmetical arrangement ; and yet much, 
as regards facihty of operation, depends upon this arrangement in many 
parts of practical mathematics, but especially in Trigonometry. 

In the present volume, therefore, both the theory and the practice of the 
science have been introduced, every particular formula being illustrated by 
examples of the numerical calculation, arranged in the proper form. This 
plan of combining practice with theory, in v/orks like the present, was 
always adopted by the earlier English writers, and it is to be regretted that 
recent authors have, in their admiration of foreign methods, departed so 
widely, hi this respect, from the example of their predecessors, dwelling so 
much as they do upon the symbols, and so httle upon the things signified. 

In addition to the practical illustration of formulas, a distinct part of the 
work is devoted to the principles of Navigation and Nautical Astronomy, 
in which will be found a very short and convenient method of clearing the 
Lunar Distance, for the purpose of ascertaining the Longitude at Sea. This 
method is probably new, although, as the analytical expression for it occurs 
during the investigation of the well-known formula of Borda, it is equally 
f" probable that it has been noticed before. 

The supplement appended to the treatise is from the pen of my valued 
and accomphshed friend, T. S. Davies, Esq. Fellow of the Royal Society 
of Edinburgh, and of the Royal Astronomical Society of London. It will 
be found to contain several new and interesting researches, which cannot 
fail to prove acceptable both to the inquiring student and to the more 
advanced analyst. 

J. R. YOUNG. 
January 1, 1833. 



CONTENTS. 



PLANE TRIGONOMETRY. 

Chap, i. Explanation to the Trigonometrical Lines. 

Article. Page. 

1. Definition of Plane Trigonometry . . . ,9 

2. On the measurement of angular magnitude . . .9 

3. Connexion between circular arcs and angles . . .9 

4. On complimentary and supplementary arcs . . .10 

5. Of the sine ........ 11 

6. Of the cosine . ; . . . . . 11 

7. Of the tangent, cotangent, secant, and cosecant . .12 

8. Of the versed sine, coversed sine, and suversed sine . . 12 

9. Abridged mode of expressing the trigonometrical lines . 12 

10. Manner of rendering trigonometrical expressions homogeneous 13' 
ib. Properties derivable from the definitions . , .14 

11. Table of the correlative values of the trigonometrical lines . 14 

Chap. n. Formulas and Rules for the solution of Plane Triangles. 

12. Right-angled triangles . . . . . .15 

13. Particulars to be observed in applying the formulas . . 15 

14. Of the arithmetical complement . . . . .15 

15. Examples . . . . . . . .16 

16. Oblique-angled triangles . . . . .17 

17. Expressions for the angles in terms of the sides . , 18 

18. Given the sines and cosines of two arcs to find the sine and 
cosine of their sum and difference . . . .19 

19. In a plane triangle the sum of any two sides is to their difterence 
as the tangent of half the sum of the opposite angles to the 
tangent of half their difference . . • . .20 

20. Formulas for determining an angle in terms of the sides . 20 

21. Remarks on these formulas . . . . .21 

22. Solution of plane triangles in general , . . .21 

23. Examples of the solution of oblique-angled triangles . . 22 

24. When two sides and the included angle are given . . 23 
ib. Examples . . . . . . . .24 

25. When the three sides are given . . . . .24 
ib. Examples . . . . . . . .26 

Chap. ni. Application of Pla7he Trigonometry to the Mensuration of 
Heights and Distances. 

25. Various problems concerning heights and distances . . 26 
1* 



d CONTENTS. 

Article. Page. 

Chap. iv. Investigation of Trigonometrical Formulas . . 32 

26. Formulas for the sum and difference of two unequal arcs . 32 

27. Proof that the angles A, B, C, of a plane triangle have this re- 
markable property, viz. 

tan. A -\- tan. B -f- tan. C = tan. A tan. B tan. C . .33 

28. Formulas for multiple arcs . . , . .34 

29. Investigation of De Moivre's Formula . . . .34 

30. General expressions for sin. n A and cos. n A deduced from De 

Moivre's formula . . . . . . .36 

ib. Various formulas for double arcs . . . .37 

31. Formulas for half arcs . . . . . .38 

32. Formulas involving the half sum and half difference of two arcs 39 

33. To express the sine and cosine of a real arc, by means of 
imaginary exponentials . . . . . .39 

ib. To develop sin." x and cos." x in terms of the sine and cosine 

of the multiples of .-c . . . . . .40 



FART II. 

SPHERICAL TRIGONOMETRY. 

Chap. I. On the Sphere. 

34. Definitions . . . , . . . .42 

35. The section of a sphere cut by a plane must be a circle . 42 

36. The circumferences of two great circles bisect each other . 43 
ib. A spherical angle is measured by the plane angle formed by 

tangents to the sides of the former, drawn through its vertex . 44 

37. The great circle distance between the poles of two intersecting 
great circles measures their angle of intersection . . 44 

ib. Every great circle passing through the poles of another is at 

right-angles to it . . . . . . .44 

38. Any side of a spherical triangle is less than the sum of the 
other two . . . . . . . .44 

39. The three sides are together less than a whole circumference . 45 

40. Characteristic property of the polar triangle . . .45 
4L Characteristic property of the supplemental triangle . . 45 

42. The angles of every spherical triangle are together greater than 
two right angles and less than six . . , ,46 

43. The arc of a great circle is the shortest line that can be drawn 

on the sphere from one point to another . . .46 

Chap. ii. Investigation of Formulas and Rules for the solution of 

Spherical Triangles. 

44. The fundamental equations of Spherical Trigonometry . 47 

45. Deduction of geometrical properties from these equations . 47 

46. Relation between the sides and opposite angles . . 49 

47. Formulas for the angles in terms of the sides . . .49 

48. Formulas involving only a, b, A, G . . . .50 

49. Formulas for the sides in terms of the angles ; . .50 

50. To convert any formula involving sides and angles into 
another similarly involving angles and sides . . .51 

51. Nanier's Analogies ,.•... 51 



CONTENTS. 



Chap. in. Solutions of the different cases of Spherical Triangles. 



Article. 

52. Formulas for right-angled triangles ; Napier's rules 

53. Examples .... 

54. Solutions of quadrantal triangles . 

55. Solution of oblique-angled triangles 

56. When the three sides are given 

57. When the three angles are given 

58. When two sides and the included angle are given 

59. When two angles and the interjacent side are given 

60. When two sides and an opposite angle are given 

61. When two angles and an opposite side are given 



Page. 

. 52 

. 54 

. 55 

. 57 

. 58 

. 59 

. 59 

. 61 

. 62 

• 64 



APPLICATION OF TRIGONOMETRY TO NAVIGATION 
AND NAUTICAL ASTRONOMY. 



Introductory remarks .... 

Chap. i. On the Principles of Navigation. 



Definitions ...... 

On plane sailing ...... 

Examples ....*. 

On traverse sailing . , , , , 

Examples . . . • , . 

On parallel sailing . . . . : 

60. Examples ...... 

67. On middle latitude sailing .... 

67. Examples .,..•. 

68. On Mercator's sailing ..... 

69. Example of Mercator's sailing .... 

70. Rule for constructing a table of meridional parts by means of a 
table of logarithmic tangents .... 

70. Examples ....... 

Chap. n. Application of Spherical Trigonometry to Astronomical 
Problems. 



66 



66 

, 68 
, 69 
, 70 
, 71 
, 73 
, 74 
. 74 
, 76 
, 76 
78-9 

78 
79 



80 



71. Description of the lines of the celestial sphere 

72. Various problems in practical astronomy 

Chap. ni. On Nautical Astronomy. 

73. Introductory remarks . 

74. Of altitude 

75. Of the dip of the horizon 

76. Of semidiameter 

77. Of refraction . 
77. Of parallax 

77. Examples of the foregoing corrections 

78. To determine the latitude at sea from the meridian altitude of 
any celestial object .... 

79. To determine the latitude by double altitudes 

80. On finding the longitude by lunar observations 

80. Three methods of deducing the true from the observed distance 102 

81. On the variation of the compass ... . 105 



89 

9a 

90 
90 
91 
92 
93 

95 

98 
100 



» CONTENTS. 

PART IV. 

MISCELLANEOUS TRIGONOMETRICAL INaUIRIES. 
Article. Page. 

82. Introductory remarks ; . . . . = 107 

83. Solutions of certain cases of plane triangles, and determination 

of the trigonometrical lines of small arcs . . 108 

84. Investigation of expressions for the surface of a spherical tri- 
angle, and for the spherical excess . . . Ill 

85. Miscellaneous formulas involving the sides and angles of a 
spherical triangle . . . . , .116 

86. Additional formulas .... . 117 

87. On the relations between the corresponding variations of the 

^ parts of a triangle .... . 118 

SUPPLEMENT. 

Chap. i. On Spherical Geometry .... 121 

Chap. n. On Associated Triangles .... 126 

Chap. m. On the Spherical Excess .... 136 

Notes . . . . , . . .146 



PART I. 

ELEMENTS OF PLANE TRIGONOMETRY. 



CHAPTER L 

EXPLANATION OF THE TRIGONOMETRICAL LINES. 

{Article 1.) Plane Trigonometry is that branch of pure mathema- 
tics of which the primary object is to determine the several parts of a 
plane triangle from having certain other dependent parts given. 

By the parts of a plane triangle we mean these six things, viz. the 
three sides and the three angles, and if any three of these six be given, 
provided only that a side be among them, the other three may always be 
determined either by geometrical construction, as shown in the Elements 
of Geometry, or by numerical computation, as will be seen hereafter. 

From the foregoing definition it appears that quantities of two kinds, 
perfectly distinct from each other and admitting of no comparison, are 
concerned in Trigonometry, viz. straight lines and angles. 

By means of certain happy contrivances, however, the whole business 
of trigonometry, and, indeed, the general theory of angular magnitude 
is conducted by help of linear quantities only; the angles themselves 
not entering into the computations, but certain straight lines dependent 
upon them and serving as indexes to them. 

(2.) Before we explain the nature of these trigonometrical lines, it will 
be necessary first to show how angular magnitude is measured. 

In order to this we must remark that when straight lines are submitted 
to calculation, all those which are concerned in the same inquiry must 
be measured in reference to one common standard of measure, called 
the linear unit; the choice of which unit is, however, arbitrary. Thus 
if we estimate any one of the lines concerned in any inquiry in feet, all 
must be estimated in feet, and the linear unit adopted will be a foot, 
which is represented by the numeral unit 1. Also if one of the lines is 
measured in yards all must be measured in yards, the linear unit being 
then a yard, which, as before, is represented in the calculation by the 
numeral unit 1. As far as the accurate representation of the lines are 
concerned, it is obviously a matter of indifference what length be as- 
sumed for the linear unit, for the leng:th of any line will always be 
expressed numerically by that number which denotes the units it contains, 
but, for the purpose of facilitating computation, some scales of measure 
are often preferable to others. 

(3.) Let now BAG be any angle concerned in any inquiry ; then 
having chosen the linear unit AB, describe the 
circumference BCD about the centre A. The 
arc BC may be taken for the measure or 
representative of the angular magnitude CAB : 
for let there be any other angle B' AC about 
the same centre A ; then we know, by Geometry, 
that the angle BAG is to the angle B'AC' as 
the intercepted arc BC to the intercepted arc 

B 




10 PLANE TRIGONOMETRY. 

B'C, (Geometry, p. 103); hence, as the intercepted arcs always vary as 
the angles, the former may, obviously, be taken to represent the latter. 

It is usual to consider the circumference of every circle to consist 
of 360 equal parts, called degrees of that circle; an arc consisting of 
any number of these, 24 for instance, is called an arc of 24 degrees, and 
represented for brevity thus, 24° ; moreover each degree is supposed to 
consist of 60 equal parts, called minutes^ and each minute of 60 equal 
parts called seconds. To express any number of minutes, we mark one 
accent over the number, and to express seconds we mark two ; thus, 
24° 16' 26", is 24 degrees 16 minutes 26 seconds. What we say of 
circular arcs applies equally to the angles which they measure, so that 
we call that an angle of 20° whose sides include an arc of 20° or the 
eighteenth part of a whole circumference. 

Let us now speak of the trigonometrical lines before adverted to, and 
which are introduced for the purpose of reducing the entire theory of 
angular magnitude to the investigation of linear quantities only ; we 
must, first, however, mention one or two further particulars respecting 
the arcs to which these lines refer. 

(4.) The arc CD which must be added to BC to make up a quad- 
rant, or 90° is called the complement of the arc BC ; and every arc will 
have a complement, even those which are themselves greater than 90°, 
provided we consider the arcs measured in the direction BCD, &c. as 
positive, and those measured in the opposite direction as negative ; thus 
the complement CD of the arc BC commences at C where BC termi- 
nates, and may be considered as generated by the motion of C, the ex- 
tremity of the radius AC, in the direction CD ; but the complement CiD 
of the arc BCi, commencing in like manner at the extremity Ci of the 
proposed arc, must be generated by the motion of Ci in the opposite 
direction, and the angular magnitude RACi, will here be diminished by 
the motion of ACi, in generating the complement; 
the complement of BACi, or of the arc BCi, is, 
therefore, with propriety considered as negative. 
Calling the arc BC, or BCi, w, the complement will 
be 90° — w ; thus the complement of 24° 16' 4" is 65° 
43' 56", and the complement of 120° 36' 10" is —30° 
3& 10'. 

The arc CBi, which must be added to BC to make 
up a semicircle, or 180°, is called the supplement of the arc BC. If the 
arc is greater than 180°, as the arc BC2, its supplement C2 Bi measured 
in the reverse direction is negative. The expression for the supple- 
ment of any arc or angle w is, therefore, 180° — w; thus the supple- 
ment of 110° 30 20' is 69° 29' 40', and the supplement of 200° W is — 
20° 25'. 

In the same manner as the complementary and supplementary arcs 
are considered as positive or negative, according to the direction in 
which they are measured, so are the arcs themselves positive or negative ; 
thus, still taking B for the commencement of the arcs, as BC is positive 
BCs will be negative. In the doctrine of triangles we consider only 

Positive angles or arcs, and the magnitudes of these are comprised 
etween w = and w = 180° ; but in the general theory of angular 
quantity, we consider both positive and negative angles, according as 
they are situated above or below the fixed line AB from which they are 
measured, as the angles CAB, Cs AB ; moreover, an angle may consist 
of any number of degrees whatever, thus if the revolving line AC set 
out from the fixed line AB and make n revolutions, and a part the 
angular magnitude generated is measured by n times 360°, plus the 
degrees in the additional part. 




TRIGONOMETRICAL LINES. 



11 




Of the Sine. 

(5.) The sine of an arc or of the angle which it measures, is the 
perpendicular, from one extremity of the arc, 
upon the diameter passing through the other 
extremity : thus CS is the sine of the arc BC ; 
Ci Si is the sine of the arc BCi ; Ca S2 is the 
sine of the arc BC2 ; Cs S3 the sine of the arc BC3, 
&c. If the proposed arc were a quadrant, or 90'^, 
the sine DA would be equal to the radius, and, 
therefore, its numerical value would be 1 ; the 
same would be the case if the arc consisted of 
three quadrants, or 270°, or indeed of any odd number of quadrants; 
for all other ar-cs the numerical value of the sine will be a proper frac- 
tion or decimal. These, it must be observed, are the trigonometrical 
values of the sines, which are estimated according, lo the scale AB — 1; 
but it should be remarked that when we know the value ol the sine of 
an arc agreeably to ihis scale, its value agreeably to any other scale is 
at once obtained by proportion; thus let R be any value assumed for 
the radius, and let us write the sine corresponding in capitals, sine ; 
then 1: sme :: R : sine = R X sine, so that the sine of an arc, corres- 
ponding to any assumed radius, is found by multiplying its trigono- 
metrical sine by that radius ; and, on the contrary, the sine according to 
any value of the radius being known, the trigonometrical sine is found 
by dividing it by that radius ; the number, in fact, which expresses the 
trigonometrical sine bemg the ratio of the geometrical line itself to the 
radius, whatever this may be. What we have said of the sine will be 
easily seen to apply to the other trigonomelncal lines. As with the arcs 
so with the sines; those which lie in opposite directions take opposite 
signs, those above the fixed line Bi B being regarded as positive, and 
those below as negative ; so that the sines in the first and second quad- 
rants are positive, those in the third and fourth negative, while in the 
fifth and sixth they are again positive, and so on. 

Every arc or angle has the same sine as its supplement ; thus if Bi Ci is 
equal to BC it is obvious that BCi will be the supplement of BC, and 
the sine CS of the latter must be equal to the sine Ci Si of the former. 

Of the Cosine. 

(6.) The cosine of an arc or angle is the sine of its complement : thus 
the cosine of the arc BC is the line Cs, which is, 
obviously, the sine of the arc DC, the complement 
of BC. As the several sines are arranged on op- 
posite sides of the diameter Bi B, so the cosines 
are arranged on opposite sides of the diameter 
DDi; those on the right of DDi being regarded 
as positive, and those opposite as negative ; hence 
m the first quadrant, the cosines are positive, in 
the second negative, in the third negative, in the 
fourth positive, and so on; the cosine of an arc is equal to the cosine 
of its supplement, but has a different sign. 

When the arc is the sine is 0, but the cosine BA is 1 ; when the 
arc is 90, the sine DA is 1, but the cosine is ; when the arc is 180° the 
sine is 0, but the cosine is Bi A = — 1 ; when the arc is 270° the sine 
Di A is — 1, but the cosine is ; and when the arc is 360° the sine is 0, 
and the cosine 1, as at first, and so on. 

It is plain that the cosine of an arc is always equal to that part of the 
radius which is intercepted between the sine of that arc and the centre. 




12 



PLANE TRIGONOMETRY. 




Thus referring to the ligure in (5) AS is equal to the cosine of BC, 
and AS2 to the cosine of BC1C2, or of BC3C2. 

Of the Tangent, Cotangent, Secant, and Cosecani. 

(7.) The tangent of an arc, and, therefore, of the angle which it 
measures, is a line drawn from one extremity of the arc, touching it at 
that extremity, and terminating in the diameter produced, drawn 
through the other extremity: thus BT is the tangent of the arcJBC 

The cotangent is the tangent of the comple- 
ment : thus Dt is the cotangent of the arc BC. 
It is easy to trace the changes which these two f^ 
lines midergo as the arc BC increases from 0, 
for which value the tangent is obviously 0, and 
the cotangent infinite. Observing the same 
rules here as for the sine and cosine, we see 
that in the first quadrant the tangent and co- 
tangent are both positive, in the second the 
tangent BTi and cotangent D^i are both ne- 
gative; in the third the tangent BTsand cotangent D^-2 are both posi- 
tive ; and in the fourth the tangent BT3 and cotangent Dfe are both 
negative, and so on ; but as we shall soon see, the signs of the tangent 
and cotangent may always be at once inferred from those of the sine 
and cosine. 

The secant of an arc is that portion of the prolonged diameter limit- 
ing the tangent, which is included between the centre and tangent ; 
and the cosecant is the secant of the complement. Thus in the last 
figure AT is the secant of the arc BC, and A^ the cosecant. 

In the four trigonometrical lines, sine, cosine, tangent and cotangent, 
we have seen that each is posited in one or other of two directly oppo- 
site directions, and that, therefore, one or other of the opposite signs -{- 
and — , prefixed to the numerical value of any such line, served to point 
out the proper direction for any particular value of the arc or angle. 
But as the secant and cosecant continually vary in direction, as well as 
in magnitude with the arc or angle, the geometrical position of either 
of these lines does not so clearly indicate to us the sign with which it 
should be represented. The proper sign, however, is always readily 
ascertained from knowing the signs of the sine and cosine, for upon 
these two lines all the others depend, as we shall shortly show. 

(8.) Besides the six trigonometrical lines now defined there are three 
others, sometimes, although but seldom, employed ; these are the versed 
sine or sagitta, the coversed sine, and the suversed sine. The versed 
sine of an arc BC (see fig. to art. 5) is the line BS between the com- 
mencement of the arc and the sine ; it is always equal to the radius 
niinus, the cosine, and, therefore, is always positive. The coversed 
sine is the versed sine of the complement, so that the coversed sine of 
BC is Ds (see fig. to art. 6 ;) also the suversed sine is the versed sine of 
the supplement. As the versed sine of any arc must be positive, it 
follows that the coversed sine and suversed sine must always be 
positive. 

(9.) The following is the way in which the trigonometrical lines, con- 
nected with any arc or angle w, are expressed in computation ; 

The sine of w is expressed thus, sin. w 
cosine of w . . cos. u 



tangent of w 

cotangent of w 

secant of w 

cosecant of w 



tan. 0) 
cot, ta 
sec. (o 

cos* 0} 



TRIGONOMETRICAL LINES. 



13 



versed sine of w . . vers, w 

coversed sine of m . . covers, w 

suversed sine of w . . suvers. w 

From knowing the numerical value of any one of these lines, those 
of all the others may be obtained ; thus, let the sine be given, then since 
the radius sine and cosine always form a right-angle triangle, of which 
the hypotenuse is the radius = 1, (see the fig. in art. 5,) we have 
COS. 0) = Vl — sin. 2 o). Again, since the triangle formed by the radius, 
sine, and cosine, is always similar to that formed by the secant, tan- 
gent, and radius, and to that formed by the cosecant, radius, and cotan- 
gent, as the student will at once see by sketching these lines for any 
arc, it follows, from the proportionality of the sides of similar tri- 
angles, that 

sin. 0) cos, 6) 1 

tan. 0) = , cot. 0) = — : — = 

cos. w sm. 0) tan. w 

1 1 



sec. w = jCosec. « = — =V 1 -l- cot. 2 w : 

cos. w sm. 0) 

and, from these expressions, we at once see that the signs of the several 
lines, as well as their numerical values, are deducible from those of the 
sine and cosine. 

Now the numerical expression for sin. w, for all values of w, from 
6) = to w == 90°, (between which limits every possible value is com- 
prised) are actually computed by methods to be hereafter explained, and 
thence the values of the other trigonometrical lines are deduced. These 
values are then arranged as in table ni, at the end, and form a table oj 
Tiaturul sines, cosines, <^c. By help of such a table we may readily find 
the values of the same lines, computed to any other radius R ; for as 
observed at (5) we shall merely have to multiply the tabular value by R. 
Writing ; therefore, for distinction sake, the words sin., cos., &c. in 
capitals, when the value of the radius is other than unity, the foregoing 

TAN. O) SIN, Oi COT. W R 

equations are the same as — =;; — =-7z^-^ , —^ — = ■ 

R COS. CO R TAN.CJ 

SEC. w R COSEC. o) R 

R ^COS. a)*'~R "^SIN. a>' 

SIN. 0) COS. a, 



and thus by substituting in any trigonometrical formula , 

R R 

&c. for sin, w, cos. w, &c. the formula will become generalized so as 
to hold good for any value of the radius whatever, 

(10.) It is obvious that when any trigonometrical formula is thus 
generalized every term in it will be the same abstract number as in 
the original formula ; whatever powers or roots of the lines enter the 
formula they will always be divided by the same powers or roots of the 
radius R. The denominators will all be removed by multiplying each 
term by the highest power of R which enters, and the result will ne- 
cessarily be a homogeneous expression ; that is, every term wiL have 
the same dimensions, or will involve as factors the same number of 
liaes. Hence, in order to generalize any trigonometrical formula, or 
to render it independent of any particular value of R, it will be necessary 
merely to introduce into the several terms such powers of R as will 
render them all of the same dimension. For example, the following 
formula, viz. 

sin. (A -|-B) = sin. A cos. B -f- sin. B COS. A ; 
in which the term on the left is of one dimension, and the terms on the 
right are each of two dimensions, will become homogeneous by intro- 
ducing the factor R into the left hand member, so that when this is the 
value of the radius instead of unity, the formula will be 
2 



14 



PLANE TPvIGOK-QMETRT. 



R sin. (A + B) = sin A cos. B -fsin. B cos. A; 
each term being the product of two lines. 

In like manner the formula cos. 4 A = 8 cos.^ A — 8 cos.2 A -}- 1, 
becomes when the radius is R instead of unity 

R3 cos. 4 A 1=^ 8 cos.'^ A — 8 R2 cos.2 A + R^ ; the powers of R being 
introduced so as to render each term of four dimensions. 

From the preceding definitions and remarks the following simple 
properties are immediately deducible, viz. 

1. The sine of an arc is equal to half the chord of twice that arc. 

2. The chord of 60° being equal to the radius (Geom. p. 122), there- 
fore, the sine of 30°, or the cosine of 60°, is equal to half the radius. 

3. Hence, from the expression for the secant at the top of the preced- 
ing page, the secant of 60° is equal to the diameter of the circle. 

. 4. The tangent of 45° is equal to the cotangent, and, therefore, to 
the radius, (see fig. to art. 7.) 

(11.) We shall terminate this introductory chapter with a table ex- 
hibiting the correlative values of the trigonometrical lines, situated in 
difl^erent quadrants ; it is readily constructed from the values of the sine 
and cosine, by help of the relations in (9), bearing in mind that an arc 
and its supplement have the same sine. 

Table of the Correlative Values of the Trigonometrical Lines. 



arc. 


sin. 




COS. 




tan. 


cot. 


sec. 


cosec. 


0° 







1 







00 


1 


c» 


0) 


+ sin. 


0) 


+ COS. 


w 


+ tan. w 


+ COt, w 


+ sec. a 


+ cosec. w 


90° 


1 









00 





00 


1 


900 + 0) 


+ C0S. 


w 


— sin. 


w 


— cot. Cd 


— tan, CO 


— cosec. CO 


+ sec. 0) 


180° 







— 1 







— 00 


— 1 


— 00 


180° + « 


— sin. 


0) 


— COS. 


Cd 


+ tan. 0) 


+ COt. CO 


— sec. u> 


— cosec. ca 


270° 


— 1 









00 





00 


— 1 


270° + CO 


— cos. 


w 


+ sin. 


OJ 


— cot. w 


— tan. CO 


+ cosec. CO 


— sec. w 


360° 







1 







00 


1 


00 



This last line is the same as the first ; and any line will, obviously, re- 
main unaltered if we add to the corresponding arc a whole circum- 
ference or any number of circumferences. If we take co negatively, we 
may extend the table as follows : 





— (ji 


— sin. CO 


+ cos. CO 


— tan. M 


— cot. CO 


90° 




+ COS. CO 


+ sin. CO 


+ cot. CO 


+ tan. CO 


180 


— CO 


+ sin. CO 


— COS. CO 


— tan. (D 


— cot. CO 


270 


0) 


— cos. CO 


— sin. CO 


+ cot. M 


+ tan. CO 


360 


CO 


— sin. CO 


+ COS. CO 


— tan. CO 


— cot. '.^ 



+ sec. 


(ji 


— cosec. 


a> 


+ cosec. 


(.0 


+ sec. 


CJ 


— sec. 


CO 


+ cosec. 


6) 


— cosec. 


CO 


— sec. 


0) 


+ sec. 


w 


— cosec. 


0) 



and by continuing this series of arcs the same values of the trigonome- 
trical lines would obviously recur as before. 

It is obvious that the cosine of a negative arc, whether less or greater 
than a quadrant, is the same as the cosine of the same arc, taken posi- 
tively ; but the sine of a negative arc, although the same in magnitude 
as that of an equal positive arc, has an opposite sign : hence, by the 
equations at (9), the sine, tangent, cotangent, and cosecant, will have 
opposite signs to those of the same arc taken positively; but the cosine 
and secant will have the same signs. 



RIGHT-ANGLED TRIANGLES. 15 



CHAPTER II. 




FORMULAS AND RULES FOR THE SOLUTION OF PLANE TRIANGLES. 

(12.) We shall now proceed to investigate rules for the solution of 
all the cases of plane triangles. 

Right-angled triangles. 
As right-angled triangles are those whose several parts are the most 
easily determined we shall consider them first. 

Le't ABC be any right-angled plane triangle, ^c 

and with AB as a radius describe the arc B«. If 
AB were unity BC would be the tangent, and AC 
the secant of the angle A ; as it is, however, these 
lines are equal to AB times the trigonometrical 
tangent and secant (5), that is, BC = AB tan A , 
AC =. AB sec. A. 

Also, bv taking the hypotenuse for the radius, . . 
we have JBC = AC sin. A, AB = AC cos. A. -^ 

These four equations, together with the geometrical property AC2 = 
AB2-|- BC2, enables us to solve every case of right-angled triangles. 

(13.) In applying these formulas, it must be remembered that the 
trigonometrical lines which they involve are according to the scale of 
radius = 1 ; they are computed and registered in the tables of natural 
sines and tangents. The tables of logarithmic-sines and. tangents are 
not, however, computed to this radius, on account of the inconvenience 
which would attend the continual use of negative indices in all the sines 
and cosines ; but they are computed to a radius of 10^ ''. Hence, in all 
formulas of trigonometry, intended for logarithmic computation, the ra- 
dius R must always be introduced, so as to make the terms homogene- 
ous ; and, although in the formulas which will be hereafter given, we 
shall but seldom encumber the expressions by actually inserting in them 
R,, and its powers, yet the computist must not fail to take account of 
them in the logarithmic process. 
Introducing R into the foregoing equations we may write them thus : 
R _tan.A R sec. A _ R _ sin. A R cos. A 

AB ~"BC~' ■AB"""^A^ ' "AC~~ BC ' AC^ AB 5 
and all these equations maybe comprehended in a single rule expressed 
as below. As the tabular radius 

: the radius in the figure 
:; any tabular line 

: the corresponding line in the figure; 
and from this it immediately follows that 
Any tabular line 

: corresponding line in the figure, 
:: any other tabular line 
: corresponding line in the figure ; 
which proportion, obviously, comprehends the former. 

It appears from this rule that when we want to find a side, we must 
begin the proportion with a given tabular line, fhat is, either with the 
tabular radius, of which the logarithm is 10, or else with the tabular 
sine, cosine, &c. of a given angle ; but when we want to find an angle 
then we must invert this proportion, beginning with a given side which 
must be made the geometrical radius, as no other tabular line but the 
radius will be given, seeing that angles are in this case unknown. 
(14.) In operating with logarithms, the logarithm of the first term of 



16 PLANE TRIGONOMETRY. 

the proportion must be subtracted from the sum of the logs, of the other 
two, to obtain the logarithm of the sought fourth term ; and thus the 
logarithmic process will consist of five lines or rows of figures. If, 
however, the first term, or that to be subtracted were 10, we might save a 
line, by adding the two other logs, together, and rejecting 10 in the in- 
dex ; when the first term is not 10 we may still save a line by the follow- 
ing artifice, viz, instead of putting down for the first term the log. given 
by the table, put down its deficiency from the number 10, which may be 
done with as much readiness as transcribing the number itself, provided 
we begin at the^ left-hand figure and subtract each in succession from 9, 
till we come to the last significant figure, which must be taken from 10 ; 
we shall thus have instead of the logarithm, what is called its arithme- 
tical complement^ which, being added in with the other two terms, reject- 
ing 10 from the index, must give the same result as if we had subtracted 
the log. of the first term from the sum of the other two. An example 
or two will fully illustrate what has now been said. 



(15.) 1, Given the angles and the base to find the perpendicular and 
hypotenuse, viz, A = 53^ 8', AB ^ 288. 

1. To find the Perpendicular BC. 

As a side is here required, we must begin 
with a tabular line ; we shall choose for sim- 
plicity the tabular radius, to save a reference 
to the table, as we know the log. of this to be 10. 
Taking then the known line AB for radius in 
the figure, we have 

Rad. . . 10 
: AB 288 24593925 

:: tan. A 53° 8' 10-1249898 




: BC 384-05 25843823; 
BC might have been found by making any other side radius, although 
not quite so easily, as we should then have had to seek out in the table 
the tabular line for the first term, corresponding to the known line AB ; 
thus if AC had been made radius, then the tabular line we should have 
commenced with, would have been that corresponding to AB, viz. the 
cosine of the angle A. If CB had been made radius we should have 
commenced with the cotangent of A, that is, the tangent of C, for such 
would be the tabular lines corresponding to B A. 

n. To find the Hypotenuse AC. 
Preserving the same radius we have, 

Rad. . . 10 
: AB 288 2-4593925 

: : sec. A 53° 8- 10-2218814 



: AC 480-036 2-6812739. 
If we had made AC radius, the proportion would have been cos. A : 
AB : : rad. : AC. By way of showing the use of the arithmetical com- 
plement, let us determine AC by this proportion 

COS. A 530 8' arith. comp. 0-2218814 
: AB 288 . . . 2-4593925 
: : Rad 10 



AC 480-036 . . 2-6812739 



EIGHT-ANGLED TRIANGLES. 



17 



2. Given the two perpendicular sides to find the hypotenuse and 
angles, viz. AB = 472, BC = 765, (see last fig.) 
I. To find the Angle A. 
We must here, agreeably to the rule, begin with a given side, say 
AB, which we shall make radius. 

AB 472 arith. comp. 7-3260580 
: Rad. . . 10 

: : BC 765 . 28836614 



: tan. A* 58° 19' 32' 10-2097194 

n. To find the Hypotenuse. 
JEiere we must begin with a tabular line ; we shall choose the radius. 



Rad. 

AB 472 

sec. A 580 19, 32^- 



10 

2-6739420 
10-2797645 



: AC 898-89 29537065. 

Or without employing the angle A we may determine AC by the for- 
mula. AC = VAB2-J-BC2. 

3. Given two sides and the included angle of 

an isosceles triangle ABC to find the other parts 

AC = BC = 288, ACB = 78° 12. 

Let the perpendicular CD be drawn, then 

since it will bisect the angle C, we shall have 

given in the right-angled triangle 

ADC , AC := 288 , ACD = 30° 6' .-.A =90° 
— 39° 6' = 50° 54 ; hence to find AD, we have 
by making AC radius, 

Rad. . . 10 

: AC 288 2.4593925 

i : COS; A 50° 54' 9-7998062 




; AD 181-635 



2-2591987 



.-. AB = 363-270. 

4. Given the base AB = 53-42, and the perpendicular BC 75-18, to 
find the hypotenuse and angles 1 

A = 54° 36^ 14', C = 35° 23' 46", AC= 72-23. 

5. Given the hypotenuse AC = 643 - 7, and the base AB = 473-8 to 
find the other parts 1 A = 42° 36' 12", C = 47° 23' 48", BC == 3587. 

6. Given the angle A = 37° 2 43", and the hypotenuse AC = 173-3 
to find the other parts 1 C = 52° 57' 17', AB 138-24, BC = 10434. 

(16.) We shall now proceed to investigate rules and formulas for 
the solution of triangles in general. 

Oblique-Angled Triangles. 

Let ABC be any plane triangle, and let us denote the angles by the 
capital letters A, B, C, at their vertices, and the sides opposite to them 
by the small letters «, h, c. 

From either vertex, as C, draw the per- 
pendicular CD to the opposite side. 

Then the sine of A to the radius b will, 
obviously, be the line C D, and the value 
of this sine in terms of the trigonometrical 
sine of the same angle to radius 1 is (art ^ 




5 A^ 



X._/ 



B 



B 



For the method of determining the angle corresponding to any tabuleir number to 
seconds, see the introductory explanation prefixed to the tables. 
2* C 



18 PLANE TRIGONOMETRY. 

5,) CD = b sin. A. In like manner the sine of B to the radius a, is the 
same line CD, whose value, therefore, in term of the trigonometrical 
sine, is CD = a sin. B ; consequently, by equating these two values of 
CD, we have a sin, B =b sin. A 

.*. — = . '" . This equation immediately furnishes us with an im- 
b sm. B ^ ■' 

portant rule, which may be expressed as follows. 

Any side of a triangle is to any other side as the sine of the angle ^ opposite 
to the former, is to the sine of the angle opposite to the latter. 

Whenever, therefore, we know two sides and an angle opposite to one 
of them, or two angles and a side opposite to one of them, the other 
three parts of the triangle may always be determined by help of this rule. 

The cosine of A, to the radius b, is the line AD ; and, therefore, AD, 
in terms of the trigonometrical cosine of A, is AD = b cos. A. In like 
manner the cosine of B to the radius BC, is BD, which, in terms of the 
trigonometrical cosine, is BD = a cos. B ; if the angle B is obtuse, as 
in the second of the above diagrams, cos. B will be negative ; hence 
whether it be acute or obtuse we shall have for the side AB the expres- 
sion c = « COS. B -j-^ COS. A ; in which the proper signs of the cosines 
are supposed to be involved in their expressions. 

If instead of drawing the perpendicular from C we had dra.wn it from 
B, it is easy to see the result we should have obtained ; for then consi- 
dering B the vertical angle instead of C, or supposing the triangle to be 
turned about till B actually becomes the vertical angle, then commenc- 
ing at the vertex, the arrangement of the angles will now be B, C, A; 
these, therefore, should respectively be substituted for C, A, B, in the 
above formula ; also the arrangement of the sides will be a, 6, c, instead 
of b, c, a, as at first, so that these letters must be replaced by the former : 
consequently, our equation will become b~c cos. K-\- a cos. C. 

If, on the contrary, Abe made the vertical angle, then the order of the 
angles will be A, B, C, and of the sides c, a, b, and these must supply 
the places, of C, A, B, and b, c, a, in the first formula, so that we shall 
then have a = b cos. C -\- c cos. B. Collecting these equations together 
we have, a = b cos. C -|- ^ cos. B^ 

b — c COS. A -j- « cos. C |> . . , . (1) ; 
c — « COS. B 4" ^ COS. A J 
and these equations contain the whole theory of plane trigonometry. 
They involve all the six parts of a triangle, the three angles, and the 
three sides ; and, as the equations are three in number, any three of the 
parts, considered as unknown quantities, may be determined, provided 
only the other three are known ; but fewer than three being given will 
not be sufficient to determine the others, as then there would be a greater 
number of unknowns than of equations. 

We must remark too that the three given quantities must not be the 
three angles simply, because the three other quantities a, b, c, severally 
enter the three terms of each equation, so that if we were to multiply 
each equation, by any assumed factor whatever, m, the values resulting 
from the elimination of A, B, C, would, obviously, be the same for ma, 
mb, mc, as for a, b, c ; thus, showing that the data are not sufficient to 
determine any triangle, bat belong equally to innumerable triangles, all, 
however, similar to each other. 

(17.) It appears then that the solutions to all the cases of plane tri- 
angles are derivable from the equations (1), under different hypothe- 
ses, as to the three imknown quantities, and we might now with but 
little trouble proceed to deduce these solutions, one after another, from 
these equations : thus suppose the three sides «, b, c, were given, then 
multiply the first equation by a, the second by b, and the third by c, we 
have a2 — ab cos. G-\-ac cos. B 



OBLIQUE-ANGLED TRIANGLES. 



19 



b2 — be COS. A-\-ba cos. C 

c2 — ac COS. B -|- Z)c COS. A ; 

and subtracting each of these from the sum of the other two, we get 

i2 + c2 — fi2 = 2 Z>c COS. A .-. COS. A = ^2 + ^^ — *^- 



-J- c2 — Z)2 = 2 ao COS. B .-, cos. B — 



^bc 

ai -{- C2 Z)2 



(2)5 



, ,o ffi2 4-j2— C2 

^2-1-^2 -c^^^ab COS. C .-. COS. C = -^^ 

and thus the values of the cosines of the required angles become known, 
and by searching in the table of natural sines and cosines we shall find 
the angles to which they belong. 

It is necessary to remark here that in almost every trigonometrical 
calculation it is advisable to conduct the operation by means of loga- 
rithms, in order to avoid lengthy and tiresome multiplications, divisions, 
and extractions ; so that it becomes a matter of consequence to express 
all our general rules and formulas in a form, adapted as much as pos- 
sible to logarithmic calculation, that is, the operations indicated by the 
formulas should be those of multiplication, division, involution, and 
evolution, and not those of addition and subtraction. 

The formulas just deduced for the angles of a tria.ngle, when the 
sides are given, do not appear in a form adapted to logarithmic compu- 
tation ; and the same would be found to be the case with the various 
other formulas directly deducible from the general equations (1); nor 
would it be easy, without the aid of other and independent properties, 
to convert these expressions into the desired form. Although, therefore, 
it is true, as we have stated above, that formulas for all the cases of 
plane trigonometry may be deduced from the equations (1), yet, on 
account of the inconvenient form these formulas assume, it becomes 
necessary for us to seek assistance from other sources. Now there ex- 
ist two general trigonometrical formulas, which may be considered as 
forming the foundation of the whole theory of angular magnitude, and 
which, in conjunction with what is laid down above, will enable us to 
deduce formulas suited to logarithmic calculation for all the cases of 
plane triangles. 

(18.) There are various ways of investigating these formulas; we 
shall adopt that which appears to us the most simple and general. 

It was given by M. Sarrus in the Annales des Mathematiques, tom. xi. 

Given the sines and cosines of two arcs or angles, to find the sine and 
cosine of their sum and difference. 

Let AM = a, and AN = a', be any two arcs of the circle, the radius 



being unity, then drawing the chord of the arc NM = 
have- from the triangle NMG right angled at G. 

MN2 = NG2 + MG2 = (Ca — CP)2 + (PM — Na)2 ; 
be written thus, chd.2 (a— a!) = (cos. a' — cos. a)2 -j- (sin 

By actually squaring the expressions in the right- 
hand member of this equation, and recollecting that 

Sin.2 a -f- C0S.2 a ::= 1, sin.2 a! -f- COS.2 a' = 1, 

we have chd.2 (a — a') = 2 — - 2 cos. a cos. a' — 2 sin. 

a sin. o! (1). 

Suppose now that o! = 0, then we have chd.2 a = 2 
— 2 cos. a.* 

As this expression is true for any arc whatever, it 
is true for the arc a — a!, so that chd.2 (a — a') = 2 — 2 cos. ( 

* This property is also proved in tlie Qcomrtry, p. 92, Scholium. 



we shall 

which may 
a — sin. a')2. 




20 PLANE TRIGONOMETRY. 

Comparing together the second members of (1) and (2) we obtain 

COS. (a — a') = cos. a COS. a' -j- sin. a sin. a' (l). 

As this is true for all values of a, a', it is true when a — a' is put for 
a', so that cos. a' — COS. a COS. (a — a') -j- sin. <i sin. (a — a')- in which 
equation, if we substitute the value of cos. (a — a') given by (i), we have 

COS. a! — cos. a2 COS. a' -[-COS. a sin. a siu. a' -{- sin. a siu. (a — a') ; 

from which, by putting for cos.s a its value 1 — sin.2 a, we get 

sin. (a — a') r= sin. a COS. a' — sin. a' COS. a . . . . (ii). 

Lastly, putting (a -j- a') for a, in the equations (I) and (II), we have, 

COS. a = COS. (a-|- a') COS. a' -[- sin. (a -[- «') sin. a'. 
sin. a = sin. {a-\-d) cos, a' — COS. (a-j- a') sin. a'. 
In order to obtain from these equations the expressions for sin. (a -|-a'), 
and cos. (a-}- a')i multiply the first by sin. a', the second by cos. a', and 
add, and we thus get,sin. {a-j- a') = sin. a cos a'-j- sin. a' cos. a . (m). 

Multiply the first by cos. a', the second by sin. a!, and subtract, and we 
get cos. (a-}- a') = cos. a COS. a' — sin. a sin. a' . . . . (iv). 

The four general formulas thus deduced may be written as follows; 

sin. (a ± a') =r sin. a COS. a' ± sin. a' cos. a > ,^ 

cos. (a ± a') = cos. a COS. a' :f sin. a sin a' 5 • • • • V -'• 

(19.) The first of these immediately furnish the two following, viz. 
sin. (a -j- a') -j- sin. (a — a') =3 sin. a COS. a' 

sin. {a -f- a') — sin. (a —a') = 2 sin. a! COS. a; from which 
sin. (a -j- a') + si^- («—«')_ sin. a. cos. a'_ tan. a 
sin. (a-f-a') — sin. (a — a') COS. a. sin. ce* tan. a', 
If, therefore, we put 

a^a' = A, a — a' = B .'. a = KA+B), a' = i (A — B), 

- „ , sin. A + sin. B tan. J (A -f B) __ 

we shall have -. ■. — — = ---^- — -~. Now we have aU 

sm. A — sm. B tan. i (A — B) 

ready seen that in any plane triangle sin. A : sin. B :'. a: b 
.'. sin. A -[-sin. B: sin. A — sin. B :; a -\- b \a — b\ 

.... . - a-\-b tan.i (A -^ B) 

consequently, from the equation above, — —, = ^^-r — ^i ; 

^ -^ ' ^ 'a — b tan. J (A — B) ' 

that is to say, in any plane triangle tJie sum of any two sides is to their 
dAfference as the tangent of half the sum of the opposite angles is to the tan- 
gent of half their difference. 

Byhelp of this rule we may determine the remaining parts of the 
triangle, when we know two sides a, b, and the included angle G ; for 
knowing C we know also i (A -f B) = i (180° — C) ; and -| (A — B) 
is determined by this rule ; therefore, as the half sum added to the half 
difference of two quantities gives the greater, and subtracted gives the 
less ; we thence readily obtain the angles A and B, and then the third 
side c, by (16.) We have thus deduced commodious rules fitted for lo- 
garithmic computation, for the solution of the first two cases of plane 
triangles : it remains to furnish a rule for the third case. 

(20.) Referring to the expression for cos. A at (17), it is plain that 
since b'2 -\- c^ = {b -\- c)2 — 2 be, and therefore, 
J2 -j- c2 — a^ =(b -\- c -\- a) (b-{- c — a) — 2bc] that expression may be 

put under the form cos. A =-^ — ■ ■ — ■ \ — — 1. 

2 be 
Now supposing the arcs a, a', in equation (A), to be equal to each 
other; and to J A, we have from the second of them 
COS. A =: C0S.2 i A — sin. ^ A 
1 = C0S.2 1 A -j- sin. 21 A 
by addition, cos. A == 2 cos.^i A — 1 
by subtractioix. cos, A = 1 — 2 sin. 2i A : 



OBLIQUE-ANGLED TRIANGLES. 21 

by substituting the first of these values in the foregoing equation, and 
putting for brevity S for the sum of the three sides of the triangle, we 

have cos. i A = J ^ — ^^^ ■'' .... (1). 

We can just as readily obtain a second formula by means of the other 
expression for cos. A ; for substituting it in equation (2), art. (17), we 

have 2 sid. ^A=l — ~-~ = si: ' = 

2 be 2 be 

_ a2 — {b— c)2 _ {a-{-b—c) (a— 6-|-c) 
— 2Tc ~ 2bc » 

consequently, sm. J A = J — j-^ . . . (2) ; 

and by dividing this expression by the former we get a third forjnulaj 

(21.) We thus have three distinct formulas for the determination of 
the angles of a triangle when the three sides are given, and all of them 
are adapted to logarithmic computation. It is not, however, always a 
matter of indifference which of these formulas we employ, as in certain 
cases one may be preferable to another. Thus, if we knew beforehand, 
or could foresee that the sought angle i- A would be very nearly equal 
to 90°, then it would be improper to employ the formula (2), because 
we should be very likely to commit error in taking out the angle, seeing 
that for an angle very near 90° the seven first decimals in the sine 
coincide with those in the sines of several other angles in its vicinity, 
or which differ each from the proposed angle by only a few seconds. 

If the logarithmic tables, which we employ, are calculated to seconds, 
as the large tables, of Taylor or of Bagay, then the sought angle when 
near 90°, may be accurately determined to the nearest second, either 
from its cosine or from its tangent, as the values of these trigonometrical 
lines, at this part of the table differ considerably from-Qiicli other, even 
when the arcs are nearly equal. But if the table employed is not calcu- 
lated to seconds, then the sought angle, when near 90°, should be deter- 
mined from its cosine, and not from its tangent ; because in approaching 
to 90° the tangents increase by very unequal differences, and, as, in propor- 
tioning for the seconds, we proceed on the supposition that the tangents 
increase equally through 60", we shall be in danger of committing error 
in thus determining the seconds. As the cosines decrease more regularly 
towards the extremity of the quadrant than the tangents increase, it 
will, therefore, be safest to determine such arcs from their cosines. 

When the sought aiigle is very small it will be best to determine it 
from its sine ; although the tangent may be used with safety. 
Solution of Plane Triangles in general. 

(22.) We shall now proceed to apply the rules and formulas which 
we have just investigated to the several cases of plane triangles, repeat- 
ing the rule at the head of each case. 

CASE 1. When a side and its opposite angle are among the given 
parts. 

RULE, — Sine of given angle, rule 2. Also, any given side, 

: its opposite side : sine of its opposite angle 

:: sine of any other angle :: any other side 
: its opposite side. : sine of its opposite angle. 

As the same sine belongs both to an angle and to its supplement, it may 
seem doubtful in determining an angle of a triangle from its sine, whe- 
ther to take the acute angle given by the tables or the obtuse angle 
which is its supplement. 



22 PLANE TPJGONOMETKY. 

The following precepts will remove all doubt on this point. 

1. If the given angle is obtuse the sought angle must be acute. This 
is obvious, because a triangle cannot have two obtuse angles. 

2. If the given angle be acute, and the side opposite to it greater than 
the side opposite to the sought angle, this must be acute ; for the greater 
angle must be opposite to the greater side. 

3. But when the side opposite to the given angle is less than that 
opposite to the sought angle, this may be either acute or obtuse, so that 
two triangles exist under the proposed conditions, and the problem in 
question admits, therefore, of two solutions. The annexed diagram 
shows that with two given sides AC, CB, and 
the acute angle A, opposite to one of them, we 
may alwaj's construct two triangles, ABC, AB'C ; 
where the angle B, opposite to the other given 
side in the one triangle, will be the supplement of 
the corresponding angle B' in the other, provided . ^ 
CB is less than CA. -^ 



(23.) 1. In the triangle ABC are given AB=137, AC=153, B=78° 13 , 
to find the remaining parts. 

I, To find the Angle C. 

As AC . 153 . arith. comp. 7-8153086 

: sin. B . 78° 1^ . . 99907502 

::AB . . 137 . . . 21367206 




: sin. C . 61° 13' 47' . . 99427794. 

The obtuse angle, which is the supplement of this, is not admissible, 
because the side opposite to the given angle is greater than the side 
opposite the required one. 

II. To find the side CB. 

The angle A is equal to 180=^ — (B-f C) =180° — 139° 26' 47' = 
40° 33 13" ; therefore, 

As sin. B 78° 13' arith. comp. 0-0092498 

: AC 153 . . . 2- 18469 14 

: : sin. A 40° 33' 13" . . . 9-8130198 



: CB 101-617 . . 20069610. 

2. In the plane triangle ABC are given AC =216, CB = 117, and 
A = 22°3T, to find the rest. 

I. To find the Angle B. 

arith. comp. 7-9318141 

. 9-5849685 

2-3344538 



AsBC 


117 


: sin. A 


22° 3? 


:: AC 


216 , 



: sin. B 45° 13' 55" or 134° 46' 5" 9-8512364. 

The angle B is, in this example, ambiguous, because the side oppo- 
site the given angle is less than that opposite the required one. 

n. To find the third side AB. 

The angle C is equal to 180° — (A + B) = 112o9' 5", provided we 
take B acute ; therefore. 



OBLIQUE-ANGLED TRIANGLES. 23 

As sin. A 22° 3? . arith. comp. 0-4150315 

• BC 117 . . . . 20681859 

:':sin. C 112° 9' &' . . . 9-9667005 



AB 281 . . . . 2-4499179 



In each of the foregoing examples where two sides, and an angle op- 
posite to one, are given, we have found it necessary to find the angle 
opposite to the other given side before we could apply the rule to the 
determination of the third side ; so that the determination of this third 
side requires two proportions, and there is no logarithmic me; hod 
which will lead us to it by a shorter process. It may, however, be de- 

duced directly from the formula at (17), viz. cos. A = -—^^, -'> 



which gives c — AB = b cos. A ± Va^ — h2 sin.2 A ; 
which expression is, however, not adapted to logarithmic computation. 
3. In the plane triangle ABC are given 

A = 44° 13' 24", B = 79° 46' 38", AB = 368, 



I. To find the side AC. 

The angle C is equal to 180° — (A-f-B) = 55° 59' 58", therefore, 

As sin. C 55° 59' 58" arith. comp. 0-0814286 

: AB 368 . . . 2-5658478 

: : sin. B 79° 46' 38" . . 9*9930503 



: AG 436-844 . . 26403267 

II. To find the side CB. 

As sin. C 55° 59' 58" arith, comp. 00814286 

: AB 368 . . . 2-5658478 

: : sin A 44° 13' 24" . . 98435174 



: CB 309-595 . . 24907938 

4. in the plane triangle ABC are given AB = 408 yards, A = 74° 14', 
B = 49° 23' ; to find the other two sides, 

AC = 371-9 yards and BC = 41602 yards. 

5. In the plane triangle ABC are given AB =z 408 yards, A = 58° ?, 
B = 22° 37' ; to find the other two sides, 

AC = 158-98 yards and BC =: 351-03 yards. 

6. In the plane triangle ABC are given AB = 318, BC = 195, A = 
32° 40 ; to find the angle C, C = 61° 40 3", or 118° 19' 57". 

Cass ii. (24.) When two sides and the included angle are given. 
RULE, (art 19.) 
As the sum of the two given sides, 
: their difierence, 

: : tangent of half the sum of the opposite angles 
: tangent of half their difference. m 

Having thus found the half difference of the unknown angles, we 
obtain the angles themselves, by first adding and then subtracting this 
half difference from the half sum. The angles being thus known, as 
well as two sides, the third side is found by the first case. 

The student will find a more compendious method of solution for 
this case in Prob. i., Part iv. ; but the rule here given will be more 
easily remembered. 



24 PLANE TRIGONOMETRY. 

EXAMPLES. 

1. In the triangle ABC are given AB = 137, AC = 153, A = 40° 3^ 
12" ; to find the other parts. 

I. To find the other two Angles. 
The sum of the other two angles is (B -j- C) = 180°— A= 139° 26' 48', 
therefore 

AsAB + AC 290 . arith. comp. 7-5376020 

: AC -AB 16 . . 1-2041200 

:: tan. KB + C) 69° 43' 24" . . 104324460 

: tan. i (B -^ C) 8° 29' 37' . 9-1741680 

By adding 78° 13' l''r- greater angle B. 

By subtracting 61° 13' 47' = less angle C. 

n. To find the third Side BC. 
As sin. B 78° 13' 1" . arith. comp. 0-0092493 

: AC 153 . . . 21846914 

:: sin. A 40° 33' 12' . . . 98130173 



: BC 101-616 . , . 2-0069580. 

2. In the triangle ABC are given AC = 378, BC = 526, 
C =32° 18' 26" ; to find the other parts. 

I. To find the Angles. 
The sum of the angles A, B is (A + B) = 180° — 32° 18' 26" = 147° 
41' 34". 

AsAC + BC 904 . arith. comp. 70438316 

: AC-^BC 148 . . 21702617 

:: tan. I (A + B)73° 50 47" . . 105381278 



: tan. J ( A -^ B) 29° 29' 34' . 9-752221 1 



By adding, 103° 10' 21" = greater angle B. 
By subtracting, 44° 31' 13" = less angle A. 

n.'i To find the side AB. 
As sin. A 44° 31' 13" . arith. comp. 0-1541818 
: BC 526 . . . 2-7209857 

:: sin. C 32° 18- 26" . . . 9" 7279 143 



: AB 400-942 . . . 2.6030818. 

If -we wish to obtain the third side of the triangle immediately, with-^ 
out first finding the angle, we may do so by means of the formula at 
(17), adverted to in the scholium to last case ; but as the computation will 
not be adapted to logarithms, it will in general be the shortest method 
to proceed as above, by two proportions. 

3. In the triangle ABC are given AB = 1637, AC = 2065, 
A=: 132° 7' 12" ; to determine the remaining parts. 

B = 26° 52' 421", c = 21°0' 5f" , BC = 3387-974. 

4. In the triangle ABC are given AB = 1686, BC = 960, 
B = 128° 4' ; to find the rest. 

A = 18° 21' 20', C = 33° 34 40', AC =. 2400-364. 
Case m. (25.) When the three sides are given. 
A rule for this case, easy to be remem.bered, may be deduced 




OBLIQUE-ANGLED TRIANGLES. 25 

from the following simple geometrical investiga- 
tion. 

Take the longest side AB of the triangle for 
base, and demit upon it the perpendicular CD 
from the vertex, which will necessarily fall 
within the base. With centre C and radius CA 
equal to the longer of the two sides AC, CB, de- 
scribe a circLe, and produce the sides AB, BC, to 
meet the circumference ; then it is plain that 

GB = AC4- CB, BF = AC — CB, BE = AD — DB. 
Now (Geom. Prop. 24, book 6). 

GB-BF = AB-BE .-. AB- (AD — DB) = (AC -f CB) (AC — CB) 
.-. AB : AG -f CB :: AC — CB : AD — DB hence the following rule. 

RULE I. 

Consider the longest side of the triangle as the base, and demit upon 
it a perpendicular from the opposite vertex, dividing the base into two 
segments; then say. As the base, 

: the sum of the other tAvo sides, 
:: the difference of those sides 
: the difference of the segments of the base. 
Having thus the sum and difference of the segments, each segment 
becomes known, and, therefore, in each of the two right-angled triangles 
into which the proposed is divided, there will be known the base and 
hypotenuse, and this is enough to determine all the other parts. 

RULE II. (art. 20.) 
coMA^J J^/^f-") , s.n. JA= J S-c)(i S- 6) 

tan'iA=JM^(|#=i) ' 

^l i S (i S — «) 

Both these rules are adapted to logarithmic computation, and this 
last is much the shortest ; when, however, the three sides are small 
numbers, it will be best to operate without logarithms, by means of the 

formula (20), cos. A = ^— ^ ■ — ^-j — ! ^ — 1. 

2 be 
In applying the logarithmic formulas in Rule 2 to the determination 
of any particular angle, it will generally be best, when this angle is op- 
posite to the longest side of the triangle, to use the first formula, and 
when it is opposite to the shortest side to use the second; the third may 
be used when the required angle is opposite to the mean side. If two 
sides of the triangle are equal, then, of course, neither of these formu- 
las will be used, as the unknown parts will be more readily found as in 
Example 3, p. 17. 

EXAMPLES. 

1. The three sides of the triangle ABC are AB = 1637, AC = 2065, 
BG = 3387-974; required the angle A. 
a = 3387-974 

* = 2065 arith. comp. 6-6850799 
c = 1637 arith. comp. 6-7859513 

2)7089-974 

h S = 3544-987 . 3-5496146 

lS — a= 157-013 . 2- 1959356 
2)19-2165814 

COS. i A = 6&0 3' 36" . 9.6082907 

/. A =-- 132° 7' 12. ^ 

3 D 



26 PLANE TRIGONOMETRY. 

2. The three sides of the triangle ABC are AB = 98, 
BC = 95-12, AC = 162-34; to determine the angle A. 
Using the third form ula in the second rule, we have 

a = 95-12 

I = 162-34 



2)355-46 





S = 177-73 arith. comp. 7-7502393 


^s- 


— a = 82-61 arith. comp. 8-0829674 


*s 


-&r= 15-39 . 11872386 


iS 


- c = 79-73 1-9016218 



2)18-9220671 



tan. i A=16° ? 26|" . 9-4610335 

.-. A = 32° 14 531". 
3. In the triangle ABC are given AC = 6, AB = 5.523, 
BC = 1-372 ; required the angle A. 
Applying the second formula to this example we have 
a = 1-372 

b^ 6 arith. comp. 9-2218487 

c =2 5-523 arith. comp. 9-2578250 



2)12-895 



hS — b= -4475 • 1-6577930 

A S — c = -9245 • • 1-9659069 



2)18-0963736 



sin. I A = 6° 24 55" . 9-0481868 

.-. A = 12^49' 50". 

4. The three sides of a plane triangle are AB = 137, AC = 153, 
BC —101-616; required the three angles, 

A = 40° 33' 12", B = 78° 13' 1", C = 61° 13' 47". 

5. The three sides of a plane triangle are AB = 1686, BC = 960, 
AC = 2400-364: required the angle B, 

B = 128° 4'. 

6. Required the angles when the sides are 4, 5, and 6. 

The angles are 41° 24 35", 55° 46' 16", and 82° 49' 9". 



CHAPTER III. 

APPLICATION aP PLANE TRIGONOMETRY TO THE MENSURATION OP 
HEIGHTS AND DISTANCES. 

PROBLEM. 1. 

A person on one side of a river observes an 
obelisk on the opposite side, and, being desirous 
to ascertain its height, he took with a quadrant 
the angle B = 55° 54', which the obelisk subtend- 
ed at the place where he stood, then going back ^ 

the distance BA = 100 feet, he again measured ^'^^ b" 

the subtended angle, and found it to be A = 33° 20 ; what was the 

height of the obelisk 1 




HEIGHTS AND "DISTANCES. 27 

In the triangle of ABC are given the angle A = 33° 20, the angle 
ACB = 55° 54' — 33° 20 = 22° 34', and the side AB ; and, therefore 
EC maybe found by Case i. of oblique angled triangles. Again, in the 
triangle BCD, we shall have given the side BC, and the angle B to find 
CD, which belongs to Case i. of right angled triangles. 

The actual computation, however, will be shortened by combining 
these two rules in a single formula, thus for the first 

AR sin A 
BC = ,^'^ , and from the second CD = BC sin CBD 
sm. ACB, ' 

^^ AB sin. A sin. CBD 

■ sin. ACB 

sin. ACB 22° 34' arith. comp. 0-4159424 

sin. A 330 20 . 9-7399748 

sin. CBD 550 54 . 99186620 

AB 100 . 2 



CD 118-57* 2-0739792. 

The problem may be solved still more readily as follows. 
If we take CD for radius, DB will be the tangent of the angle DCB, 
and DA the tangent of DC A, therefore, AB is the difference of those 
tangents j but by referring to the table of natural tangents, we find that 
to radius 1 

nat. tan. 56° 40 = 1-5204261 
nat. tan, 34° 6' = 6770509 



difference = -8433752 
.-. -8433752 : 1 : : 100 : 11857, as before. 

PROBLEM II. 

A person at A wishes to know his distance from an 
inaccessible object at C, but he has no instrument for 
taking angles. He, therefore, sets up a staff at A, from 
•which he measures the distance AA = 60 feet, so that 
when he stands at A the staff and the object appear in 
the same straight line AC ; he in like manner, mea- 
sures another distance BB' = 86 feet, from a second sta- 
tion B, 38 feet from the former A, and he finds the dia- K^ 
gonal distances AB', BA, to be respectively 97 feet and 81 feet. From 
these data it is required to determine the distance of A from the object C. 
All the three sides of the triangle A'AB are given, therefore to find 
the angle A'AB we have, by using the first formula at (25), 

A'B = 81 

A'A = 60 arith. comp. 8-2218487 

AB = 38 arith. comp. 8-4202164 

2)179 

JS= 89-5 . 1-9518230 
i S — A'B= 8-5 , 0-9294189 




2)19-5233070 



cos. J A AB = 54° 41' 56" 9.7616535 

,-, A'AB = 109° 23' 52" .-. CAB = 70° 3&8". 

• To the height of the object thus detPrminpH thp heiglit of fhP observer's eye, or of 
the instrument, must be addfd. 



28 PLANE TEIGONOMETRT. 

Agaiiij by applying the same formula to the triangle B'BA, we have 
B'A = 97 

B'B = 86 arith. comp. 8-0655015 
AB ^ 38 arith. comp. 8-4202164 

2)221 

110-5 . 2-0433623 

13-5 . 11303338 



2)19-6594140 

COS. I B'BA =z 47° 29' 50' . 9-8297070 

.-. B'BA = 94° 59' 40' .-. CBA = 85° 20' 
.-. C = 180 — (CAB + CBA) = 24°23' 40". 
Consequently, in the triangle ABC, we have all the angles and one side 
AB given ; hence, by Case i. 

sin. C 24° 23' 40' arith. comp. 0-3840330 
: AB 38 . . . 1-5797836 

:: sin. B 85° 20' . 99983479 



: AC 91-657 . .. 19621645. 

PROBLEM m. 

At the top of a castle, which stood on a hill near the sea-shore, the 
angle of depression HTS, of a ship at anchor, was observed to be 4° 52" ; 
at the bottom of the castle the angle of depression OBS was 4° 2*. 
Required the horizontal distance AS of the vessel, and the height of 
the hill above the level of the sea, the height of the castle being 60 
feet. 

As TH, BO, are parallel to AS, we have TSA = 4o 52', and BSA 
= 4° 2'. Bearing this in mind we have 

InATSB, BT-skTTSB 



In A BSA, AS = SB cos. BSA, AB = SB sin. BSA 

. -, _ BT sin. ATS cos. BSA . _ ^ BT sin. ATS sin. BSA 

sm. 1 SB sm. 1 oB 

hence the logarithmic operation will be 

sin. TSB 0° 50 arith. comp. 1-8373192 . . . 1-8373192 

sin. ATS 85° 8' . . 9-9984315 . . . 9-9984315 

cos. BSA 4° ^ . . 9-9989230 sin BSA . 8-8471827 

BT 60 . . . 1-7781513 . . . 1-7781513 



AS 4100-4 . . 3-S128250, AB 289-12 . 2-4610847. 

PROBLEM IV. 

The distances of three objects A, B, C, from 
each other, are as follow, viz. AB = 462 yards, 
AC = 328 yards, and BC = 297 yards ; a per- 
son at D, wishing to know his distance from 
each object, takes the angle ADB, and finds it 
to be 34° 16' 21" ; it is required to determine ^ 
DA, DC, and DB. 

As the three sides of the triangle ABC are given we may find the 
angle CAB, and, consequently, the supplemental angle DAB, so that 




HEIGHTS AND DISTANCES. 29 

we shall have in the triangle DAB the two angles D, A and the side 
AB to find the rest. The computation will, therefore, be as follows, 

I. To find the angle CAB. 

BC = 297 

AC = 328 arith. comp. 7-4841262 

AB =462 arith. comp. 7-3353580 

2)1087 



543-5 
215-5 


2-3334473 


815 


2-9111576 



2)190640891 

sin. i CAB = 19° 54 14' . 95320445 

. DAB = 180° — 39-^ 48' 28' = 140° 11' 32' .-. DBA = 15° 32' 7' 

II. To find AD. 
As sin. D 24° 16' 21" arith. comp. 0-3860770 
: AB 462 . . 26646420 

:: sin. B 15° 32' 7" . . 94278619 



: AD 30101 . . 2-4785809 

.-. DC = DA + AC = 629101 yards. 

m. TofindBTi. 

As sin. D 24° 16' 21" arith. comp. 0-3860770 

: AB 462 . . 26646420 

:; sin. A 140° 11' 32" . 98063252 



: BD 719-522 . . 28570442 

Hence we have the three distances, viz. DA = 30101, DC = 629-101, 
DB = 719 522. 

PROBLEM V. 

Suppose that from the top of a mountain, three miles high, the angle 
of depression of the remotest visible point of the earth's surface is taken 
and found to be 2° 13' 27" ; it is required thence to determine the dia- 
meter of the earth, supposing it to be a perfect sphere. 

Let O be the centre of the earth, BA the mountain, ^ 

AC the visual ray or line touching the earth's surface b^^ "™ 

in C. Draw the tangent BD, and join OD, OC ; then j^ //^ 
the angle of depression EAC being given, we have / 1/ ys 

also the angle BAD, the complement of it, equal to / r ] 

87° 46' 33". Also since the tangents BD, CD, are I ^ / 
equal, (Geom. p. 106,) we have the angle BOD = V / 

DOC = i comp. A = 1° 6' 49 J", and, therefore, \ ^ 

BDO = 88° 53' 161". 

Now in the right-angled triangle ABD we have BD = AB tan. A; 

and in the right-angled triangle OBD, OB = BD tan. BDO ; hence by 

substitution, OB = AB tan. A tan. BDO ; the computation is, therefore, 

as follows : AB = 3 . . 4771213 

tan. A 87° 46' 33" . 11-4107381 

tan. BDO 88° 53' 16^" . 117119309 



OB 3979-15 . . 3-5997903; 

hence the diameter is 7958-3 miles. 
3* 



30 



PLANE TRIGONOMETRTo 




PROBLEM VI. 

Given the distances between three objects A, B, C, and the angles 
subtended by these distances at a point D in the same plane with them ; 
to determine the distance of D from each object. 

Let a circle be described about the triangle ADB, and join AE, EB, 
then will the angles ABE, BAE, be respectively equal 
to the given angles ADE, BDE, (Geom.p. 52) ; thus all 
the angles of the triangle AEB are known, as also the 
side AB ; we may find, therefore, the remaining sides 
AE, EB. Again, the sides of the triangle ABC being 
known, we may find the angle BAG ; hence the angle 
CAE becomes known, so that in the triangle CAE we 
shall have the two sides AE, AC, and the included 
angle given, from which we may find the angle AEC 
in fig. 1, or the angle ACE in fig. 2, and thence its 
supplement AED or ACD ; this with the given side 
AE and angle ADE, in the first figure, or with the 
given side AC and angle ADC in the second, will 
enable us to find AD, one of the required lines, and 
thence DC and DB, the other two. 

Or the solution may be conducted more analytically as follows. 

Put X for the angle DAC, and re* for the angle DBC ; also call the 
given angles ADC, BDC, a and a', then a^ b, c, representing as usual the 
sides opposite to A, B, C, we have 

sin. a b sin. a' a /'^^ . sin. asin. rtf b 

sin. X ~~ DC ' sin. x' ~ DC ' ' ' ' ^ J ' ' sin a' sin. x ~ a 
.-. a sin. a sin. xl = b sin. a' sin. x . . . . (2). 

This is one equation between the unknown quantities x, xl. Another 
is easily obtained ; for since the four angles of the quadrilateral ADCB 
make up four right angles or 360°, we have x-\-x' -[- a-\- a' -{- ACD -j- 
BCD = 360° ; the two latter angles maybe considered as known, since 
in the triangle ABC the angle C is determinable from the three given 
sides ; therefore all the terms in the first member of this eqaation are 
kno-wTi except x and a^. Call the sum of these known quantities /?, and 
we shall thus have .-c' = — x, and, consequently, by substitution, equa- 
tion (2) becomes, a sin. a sin. (/? — x) — b sin. a! sin. x 
— a sin. a (sin. /? cos. x — cos. /? sin. a;) ; 
or dividing by sin. x, b sin. a! z= a sin. a (sin. /? cot. x — cos. /?) 

b sin. a! , COS. /? b sin. a' , . „ 

.'. cot. X = : : -\ : = : : — + COt. /?. 

a sm. a sm. ' sm. p a, sm. a sm. /? 

The first term of this second member may be easily calculated by 
logarithms, and this added to the natural cotangent of /? gives the nat. 
cot. of a;, and thence a;' is known from the equation x! = P — x, and CD 
from either of the equations (1). 

PROBLEM vn. 

Given the angles of elevation of an object taken at three places on 
the same horizontal straight line, together with the distances between 
the stations ; to find the height of the object and its distance from either 
station. 

Let AB be the object, and C, C, C", the three stations, 
then the triangles BCA, BC'A, BC'A, will all be right 
angled at A ; and, therefore, to radius BA, AC, AC, AC", 
will be the tangents of the angles at B, or the cotangents 
of the angles of elevation; hence putting a, a', o", for the 
angles of elevation, x for the height of the object, and a, b, 
for the distances C C, C C", we shall have AC = x cot. 

a, AC' = X cot. a'. AC" = X COt. a". 




HEIGHTS AND DISTANCES. 31 

Now if a perpendicular AP be drawn from A to C C", we shall have 
(Geom. p. 35,) from the triangle ACC' 

AC2 = AC'2 4- C' C2 — 2 C C • C'P ; and from the triangle ACC 
AC"2 = AC'24- C" C'2 + 2 C" C • C P ; that is, we shall have the two 
equations a^ cot.2 a = .t2 cot.2 d -\- a'i — 2a- C'P. 
a;2 cot.2 a" = a;2 cot.2 a' -{- &2 -{- 2 6 • C'P. in order to eliminate C'P, mul- 
tiply the first by i, the second by a, and add and we shall have 
rc2 (5 cot.2 a-{- a cot.2 a") z={a-\-h) x^ cot.2 a' -j- ab {a-{-b) 

_ I ab{a-Jrb) 

~ N 6 cot.2 a-\- a cot.2 a" — {a -\- b cot.2 a') 
If the three stations are equidistant, then a = b, and the expression 

becomes x = — — 

Vi cot.2 a -f i cot.2 a'J — cot.2 a' ' 

The height AB being thus determined, the distances of the stations 
from the object are found by multiplying this height by the cotangents 
of the angles of elevation. 

PROBLEM vm. 

Three objects A, B, and C, whose distances are AC = 8 miles, BC 
= 74- miles, and AB = 12 miles are visible from one station D, in the 
line joining A and B, at which point the line joining A and C subtends 
an angle of 107° 56' 13". Required the distances of the objects from the 
station. AD .= 5 miles, DC = 4-892 miles, DB = 7 miles. 

PROBLEM IX. 

Suppose the angle of elevation of the top of a steeple to be 40° when 
the observer's eye is level with the bottom, and that from a window 
18 feet directly above the first station, the angle of elevation is found to 
be 37° 30'. Required the height and distance of the steeple. 

Height = 210-44 feet. Distance 250-79 feet. 

PROBLEM X, 

In order to determine the horizontal distance between two remote 
objects A, B, a base line A' B' of 536 yards was measured, and then a 
flagstaff being set up at each extremity, these four angles were taken 
from them, viz. at A' the angular distance between A and B, 57° 40', 
and the angular distance between B and B', 40° 16', also at B' the an- 
gular distance between A and B, 71° 7', and the angular distance be- 
tween A' and A, 42° 22. Required the distance between the objects. 

939-52 yards. 

PROBLEM XI, 

Three objects A, B, C, are in the same straight line, and of known 
distances from each other, viz. AB = 3626 yards, and BC == 8-374 
yards, the angular distance of A, B, from a station D, where all the 
objects are visible, is 19°, and the angular distance of B, C, is 25°. 
Required the distance of each object from the place of observation. 

DA = 9-471 yards, DB = 10861, DC = 16848. 

PROBLEM XII. 

At three points in the same horizontal straight line the angles of ele- 
vation of an object was found to be 36° 50', 21° 24' and 14°, the middle 
station being 84 feet from each of the others. Required the height of 
the object. 53964 feet. 

PROBLEM Xni. 

There are three towns A, B, and C, whose distance apart are as fol- 
low : from A to B six miles ; from A to C, 22 miles ; and from B to C, 
20 miles. A messenger is despatched from B to A, and has to call at a 
town D in a direct line between A and C. Now in travelling from B 



iSa PLANE TRIGONOMETRY. 

to D, he walks uniformly at the rate of 4 miles an hour, and from D to 
A at the rate of 3 miles an hour. Supposing him to perform his jour- 
ney in 3 hours, it is required to determine the position of the town D. 
The distance of D from A is 4' 72 miles. 
The student who has the practical applications of Plane Trigono- 
metry more immediately in view, may pass over the following chapter, 
on the theory of the trigonometrical lines, and proceed to the first chaj)- 
ter of part m., which contains the application of Trigonometry to Navi- 
gation. 

CHAPTER IV. 

INVESTIGATION OF TRIGONOMETRICAL FORMULAS, 

(26.) The formulas hitherto investigated are those only which are 
immediately connected with the business of plane trigonometry, pro- 
perly so called, that is, with the solutions of the several cases of plane 
triangles. Having disposed of all these cases, we shall now proceed to 
develop the theory of the trigonometrical lines more at large, dismiss- 
ing all considerations of the sides of triangles. 

The following general expressions have already been established, viz. 
sin. (A -f- B) = sin. A cos. B -|- sin. B cos. A > /,>. 

sin. (A — B) = sin. A cos. B — sin. B cos. A ^ ' ' ^ '' 
COS. ( A -{- B) = COS. A cos. B — sin. A sin. B \ /g. 

cos. (A — B) = COS. A COS. B -j- sin. A sin. B J ' ' ^ '' 
From these equations we get 

1. By addition, 

sin. (A 4- B) + sin. ( A — B) = 2 sin. A cos. B > (o\ 
COS. (A -f B) + cos. (A— B) = 2 cos. A cos. B J ' ' ^'*^* 

2. By subtraction, 

sin. (A -|- B) — sin. ( A — B) = 2 cos. A sin. B > ,.. 
cos. (A — B) —cos. (A -f B) = 2 sin. A sin. B ^ ' * ^^'• 
It is worth while to remark here that if we make A — 60°, then since 
COS. 60° A I, (p. 14,) the first of these formulas furnish the equation 

sin. B = sin. (60° + B) — sin (60° — B) . . ( V) ; 
which is a useful expression in the work of computing tables. 

3. By multiplication, 

sin. (A 4- B) sin. (A — B) = sin.2 A cos.s B — sin.sB cos.2 A 
cos. (A + B) cos. (A— B) = cos.2 A cos.2 B — sin.2 A sin. 2 B. 

Or eliminating cos.2 A, cos.2 B. from each of these equations by means 

of the conditions sin.2 A -|- cos.2 A = 1 ; sin.2 B -f cos.2 B = 1; the 

second members of them become, respectively, 

sin.2 A — sin.2 A sin.2 B — sia.2 B -f sin.2 B SL11.2A, or sin.2 A — sia.2B ; 

and, 

1 — sin.2 B — sin.2 A -\- siii.2 A sin.2 B — sin.2 A sin.2 B, or cos.2 B sin.2 A; 

so that 

•sin.(A+B)sin(A— B)=sin.2A— SU12B— (sinAH-sin.B)(sinA— sin.B) \ 

cos(A-f B)cos(A— B)=cos2B— sin2A=(cos.B-^sin.A)(cos.B— sinA) \ ^^ 
. „ ,. . . sin. (A 4- B) sin. A cos. B -f sin. B cos. A 

4. By division, v 1 ^ _ 



sin. (A — B) sin. 
cos. (A -|- B) cos. 


A cos. 
A cos, 


B — sin. 
, B — sin. 


B cos. 
A sin. 


A 
B 


cos. (A — B) "^ cos. 
sin. (A ± B) sin. 


A cos. 
A cos. 


B -f sin. 
B ± sin. 


A sin. 
B cos. 


B 

A 


COS. (A ± B) ~ cos. 


A cos. 


B T sin- 


A sin. 


B 



The right hand number of these equations will assume other useful 
forms by dividing both numerator and denominator of each by certain 
expressions : thus, let the divisors for the first equation be 



TRIGONOMETRICAL FORMULAS. 33. 

COS. A COS. B, sin. A sin. B, sin. A cos. B ; 
those for the second, cos. A sin. B, sin. A cos. B, cos. A cos. B ; and 
those for the third the same as those for the first ; we shall then have 
sin. (A + B) tan. A + tan. B _ cot. B-|-cot . A l + cot- A tan. B" 
sin. fA — B)~'tan. A — tan. B~cot. B — cot. A"~l — cot. A tan.~^ 
cos.(A-l-B) cot. B — tan. A cot. A — tan. B 1 — tan. A tan. B 



H6)- 



cos. (A — B) cot. B -j- tan. A cot. A-+ tan. B 1 -f tan. A tan. B 
sin. (A ± B)_tan. A ± tan. B cot. B ± cot. A_ 1 ± cot. A tan. B 

COS. (A i B)"~l:f tan.Atan.B~~cot.Acot.B:Fl~cot. A :f tan. B 

The last of these immediately gives 

. . , _. tan. A 4- tan. B ^ ,. __ tan. A — tan. B"^ 
tan. ( A + B) = ^ -,' tan. ( A — B) =- — — — ■- 

^ ^ ^ 1 — tan. A tan. B ^ ^ 14-tan. Atan.B ! 

^* , -ON cot. Acot. B — 1 ,. _,^ cot. Acot.B+1 f '^^'' 

cot. (A -f B) r= -— -— -, cot. (A — B) ^ -; ^- I 

^ ^ ^ cot. B -f- cot. A ^ ^ cot. B— cot. A J 

If A = 45°, then tan. A = cot. A = 1, therefore, 

r*rr. I Tix 1 + tan. B ^ .._^ _. 1 — tan. B 

tan. (45° + B) = -—- =-, tan. (45° — B) = — = 

^ ' ^ 1 — tan. B ^ ^1-1- tan. B 

^Ar- , -ON cot. B — 1 ,,^ ^^ cot. B + 1 

cot. (45° + B) = ^ „ , » cot. (45° — B) == , t. -i 

' ^ cot. B -f- 1 cot. B — 1 

.•.tan.(45o + B)-tan. (45o_B)=^iil^lB * * ' (^) 
cot. (450 _ B) - cot. (450 + B) = ^^i^^^ • . • (9). 

Such are the most useful theorems respecting the sums and differences 
of two unequal arcs, and they may be converted into other expressions, 
involving three or more arcs by simply substituting B + C -f- D -|- &c. 
for B. We shall briefly consider the case of three arcs, or angles, 
because of a curious property belonging to them whenever they make 
up either 180° or 90°. 

Let A, B, C, be any three arcs, and consider A + B as one, then by 
equa. (1) 

sin. (A + B -f- C) =: sin. (A + B) cos. C + cos. (A + B) sin. C 

= (sin. A cos. B -j- cos. A sin. B) cos. C -f- 

(cos. A cos. B — sin. A sin. B) sin. C, 

cos.(A+B + C) = cos.(A + B)cos.C — sin.(AH-B)sin.C 

= Tcos. A cos. B — sin. A sin B) cos. C — 

(sin. A cos. B + cos. A sin. B) sin. C. 

Let now the sum of the three arcs be 180°, or, indeed, any multiple 
of 180°, then the sine of this sum will be 0, so that the first of these 
equations gives sin. A cos B cos. C -\- cos. A sin. B cos. C -f- 

cos. A cos. B sin. C = sin. A sin. B sin. C; 
dividing both sides of this equation by cos. A cos. B cos. C, we have 
sin. A sin. B sin. C _ sin. A sin. B sin. C 
COS. A COS. B ' cos. C ~ cos. A * cos. B ' cos. C' 
that is, tan. A --(- tan. B -{- tan. C = tan, A tan. B tan. C ; 
a remarkable property of the angles of a plane triangle. 

Again, let the sum of the three arcs be 90°, or any multiple thereof, 
then the cosine of this sum will be 0, so that the second general equation 
above becomes cos. A cos. B cos. C = cos. A cos. B cos. C -|- 

sin. A COS. B sin. C ~\- cos. A sin. B sin. C ; 
dividing both sides by sin. A sin. B sin. C, we have 

cot. A cot. B cot. C = cot. A ^ cot. B -f- cot- C. 
E 



34 PLANE TRIGONOMETRY. 

(28.) To deduce formulas for multiple arcs we have only to put w A 
for A -[- B in the preceding expressions. We thus get from (1) 
sin. nK — sin. A cos. {n — 1) A -j- sin. {n — 1) A cos. A 
cos. 71 A = cos.A cos. (ii — 1) A — sin. A sin. (% — 1) A; 
so that putting for n, 1, 2, 3, &c. successively, we have 
sin. A = sin. A 
sin. 2 A = 2 sin. A cos. A 

sin. 3 x^ = sin. A cos. 2 A -f sin. 2 A cos. A j* . . . (10) 
sin. 4 A == sin. A cos. 3 A -j- sin. 3 A cos. A 

&c. &c. 

cos. A = cos. A 
COS. 2 A = C0S.2 A — sin.2 A 

cos. 3 A = cos. A cos. 3 A — sin. A sin. 2 A )> . . . (11). 
COS. 4 A = cos. A COS. 3 A — sin. A sin. 3 A 
&c. &c. 

We may put the general expressions for sin. tz-A, and cos. %A, under a 
different form, by making use of the second equation in (1) and (2), 
thus putting {n — 1) A for A, and A for B, these become 

sin. (?i — 2) A = sin. {n — 1) A cos. A — sin. A cos. {n — 1) A 
COS. {n — 2) A = cos. {n — 1) A cos. A -|- sin. {n — 1) A sin. A ; 
or, by transposing, 

=r — sin. A cos. (n — 1) A -|- sin. {n — 1) A cos. A — sin. (ii — 2^ A 
= + cos. Acos. (w — 1) A -{-sin. A sin. {n — 1) A — cos. (ti — 2) A; 
adding these two equations to those above, there results 

sin. ?i A = 2 sin. {n—l) A co§. A — sin. (ti — 2) A^ <r.„> 

cos.7iA = 2cos.(7i— 1) ACOS..A— cos.(w — 2)A5 ' * * ^^^^» 
hence, sin. A = sin. A 

sin. 2 A = 2 sin. A cos. A 

sin. 3 A = 2 sin. 2 A cos. A — sin. A y . . . (13) 

sin. 4 A = 2 sin. 3 A cos. A — sin. 2 A 

&c. &c. 

cos. A = COS. A 
cos. 2 A = 2 COS. A cos. A — 1 

COS. 3 A = 2cos. 2 Acos. A — COS. A V . . . (14). 
cos. 4 A = 2 COS. 3 A cos. 2 A — cos. 2 A 
&c. &c. 

(29.) The sines and cosines of multiple arcs may also be developed 
in terms of the powers of the sine and cosine of the simple arc, by help 
of a remarkable formula, known by the name of De Moivre's formula, 
which may be easily established, as follows. 
Multiply together the two expressions, 

cos. A -f sin. A . V — 1 and cos. Ai -j- sin, Ai . V — 1 ; 
and we shall have the product, cos. A cos. Ai — sin. A sin. Ai -f- 
(cos. A sin. Ai -|- sin. A cos. Ai) V — 1 ; which, by the equations (1), 
(2), is the same as, cos. (A -\- Ai)-j- sin. (A-}- Ai) V — 1 ; 
which is of the same form as the original factors, consequently, multi- 
plying this by the new factor, cos. As -{-sin As . V — 1, we must have for 
the product cos. ( A -{- Ai -f As) -f- sin. (A -j- Ai -|- A2) V — 1, 
and thus by continually introducing a new factor, we must have 
generally 

(cos. A -f- sin. A . V — 1) cos. Ai -{- sin. Ai . V— i) (cos. As -f- 

sin. A2. V— 1) &c. = 

cos. (A -f Ai -I- As 4- &c.) + sin. ( A + Ai -f As -^ &c.) -v/ — 1. 
Suppose now that A — Ai = As = &c. then this equation will become 



TRIGONOMETRICAL FORMULAS. 



35 



(cos. A -f- sin. A . V — 1)'' = cos. n A-\- sin. n A. V — 1 
or, writing the radical with the double sign, 

(cos. A ± sin. A.V — 1) « = cos.7t A± sin. nA.V — 1 . . (15); 
nis here a whole number, but, in order to show that the formula holds 

when the exponent is a fraction, put a = — A; then by this formula, 



(cos. a ± sin. a . V — 1) '" = cos. ma ± sin. mA .V — 1 



cos. 71 A ± sin. n A .V —\ = (cos. A ± sin. A.V — 1) " ; there- 
fore, extracting the mth root of the first and last members, restoring the 



n , . . n 



value of a, we have, cos. — A ± sin. — A.V — 1 = (cos. A 
m m 

± sin. A.V — 1) '" . (16) ; which is the formula of De Moivre. 

If we take the reciprocal of each side of this question we shall have 



n 



— (cos. A i sin. A . V — 1) 



COS. — A i sin. — A . V — 1 

m m 

and if we multiply both numerator and denominator of the first member 

of this by cos. A T sin, — A.V — 1, the denominator will then 

m m 

become cos. ^ — A -\- sin. A = 1 ; hence 



cos. — A qp sin. — A. V — 1 = (cos.A ± sin.A.V — 1) "* (17): 
mm 

so that the formula (16) remains true, whether — be positive or nega- 
tive. If in (16) we make — negative, the signs ±, in the first mem- 
ber, will be inverted as here, because the sign of the sine is the same 
as that of the arc. 

It may seem to the student that there is a want of generality in the 
first members of (16) and (17), which ought to contain m values, seeing 
that the mth root appears in the second members. But this defect is 
only apparent ; for it must be remembered that while the lines sin. A, 
cos. A, in the second member have each a certain fixed value, the arcs 
A, to which these lines indifferently belong are innumerable. The first 
member involves a proposed fractional part, not of any particular one 
of these arcs, but of any one of them indifferently ; it is easy to see, 
therefore, that the first member involves a variety of values, and they 
may be shov/n to be in number m. 

We are to show here that in formula to De Moivre, viz. 



cos. A ± sin. A- V — 1 = (cos. A ± sin. A.V — 1) "* 

mm 

the first member has m values as well as the second. This fact we 
shall easily establish, by means of the property adverted to in the text, 
viz. that to any given values of the lines sin. A, cos. A, there corres- 
pond innumerable different arcs, viz. every arc in the infinite series, 

A, 2 TT -f- A, 4 TT -f- A 6 TT -j- A, &c. 
so that the first member of the above formula involves in it the follow- 
ing values, viz. cos. — A ± sin. — A.V — 1 
m, m 



n 



COS. (2 TT -I- A) i sin. (2 tt 4- A) . V — 1 

m m 



PLANE TRIGONOMETEY. 



n 



— (4:7+ A) isin. (2 77+ AY. V—l 

7n m 



COS. — (Gtt + A) ± sin. — (6 ;r -f A) . V— 1 

&c. &c. 

These values will continue different till we arrive at such a value, N, 

for one of the numerical coefficients, 1, 2, 4, 6, &c. as will render — Nw 

a multiple of 27r, when the first of the foregoing values will ob- 
viously recur, so that by continuing the series we shall merely obtain a 

repetition of the former values. Now — N tt cannot become a multiple 

m 
of 2 TT till N become equal to '•2m ; hence we shall !have expressed all 
the different values involved in the first member of De Moivre's for- 
mula, when we have continued the above series of values as far as that 
in which the numeral coefficient is 2m — 2 ; that is when we have 
written m values. Hence each member of the formula involves m 
different values. 

(30.) Let the first side of (15) be developed by the binomial theorem 
and the equation will become cos. " A ± ?i cos. " ^ Ap 

-\ — —- — '— cos. A^2 i &c. = cos. n K± sin. n K.V — 1 ; 

'Z 

■p being put for the imaginary sin. A . V — 1. 

Nov: as in any equation the imaginaries on one side are together equal 
to those on the"'otber, (Alg. p. 88,) we have by expunging all the ima- 
ginaries on both sides, the following expression for cos. n A, viz. 

n^n — 1) „ « . . . 

cos. n A=: COS. "A — ^ cos. "-^ ^ 5^2 A -f 

2 

^-^ ' .^ , COS. " -4 A sm. 4 A — &c. 

2 • 3 -4 
Id the like manner by expunging all the rational terms on each side 

of the same equation, and then dividing by V — 1, there results for 
sin. n A, sin. n K-= n cos. " — 1 A sin. A 

- " '"-P ^^-^^ cos.-^ Asm.3 A + &e. 

From these two expressions may be obtained series for the value of 
the sine and cosine of an arc in terms of the arc itself. 

For let n = — -, and sin. A = = A, then n K— — = any finite 

quantity x\ hence by these substitutions the foregoing series become 

.sm. X ■=. X —A &c. 

1-2-3 ^ 1-2-3-4-5 
by means of which we may calculate the values of the sine and cosine 
of any arc x, in parts of the radius or linear unit, when we know the 
^length of X itself, according to the same scale. The length of any arc 
in parts of the radius is easily ascertained from the known value of 
180=' or of a semicircle, in those parts, which by putting - for the semi- 
circumference to radius 1, is (see Geom. p. 139) 1:=^ 3- 14159265358979, 

&c. so that the length of an arc x degrees is -^ . t = — - . -^ . 

180 90 2 

j^s in calculating the smes and cosines x may be always taken less than 



TRTGONOMETFJCAL FORMULAS. 37 

X 

90, it follo-v\-s that -^ will be a decimal fraction ; if we call this m we 
may write the foregoing series thus, 

COS. {m • 90°) = 1 — -yf^ m2 -1 — ^^J?', m^ — &Q. 

which series are now in a form suited to immediate calculation. 

Suppose, for example, the sine and cosine of 1' are required, then, 

m = ^ .-. m ^^- =^ -0002908882, &c, .-. sin. 1' = -0002908882, 

&c. — ^ \^ (-0002908882, &c.y + &c. = -0002908882, &c. 

-— ^ (0002908882, &c.)2 &c. =: -9999999577, &e. 
1 ■ 2 

and from knowing the value of sin. 1' and cos. 1' we might compute the 
sines and cosines for every minute in the quadrant, by means of the 
formula (3), which when B = 1', becomes 

sin. (A -f- 1') = 2 sin. A cos. 1' — sin. (A — 1'), 
in which A is to be made successively equal to 1', 2', 3', &c. But we 
shall not enter into the details of this computation here, our present ob- 
ject being to deduce formulas for the sines, cosines, &c. of multiple arcs. 
From the general expressions already given for sin. nA, and cos. %A, 
those for tan. 7iA, cot. %A, &c. may be readily obtained by help of the 
equations at (9) ; we shall not, therefore, occupy the space by writing 
them down, but confine ourselves throughout the remainder of this 
article entirely to the consideration of double arcs, as formulas for these 
are in much more frequent request than for any higher multiple. The 
formulas of which we speak, may, of course, all be deduced from the 
general expressions investigated in the beginning of this article, but, 
for the sake of simplicity, we shall go nearer the first principles, and 
deduce them from the expressions in art. (26). 

Referring to the equations (1), (2), art. (26), we have when A — B, 
sin. 2 A = 2 sin. A cos. A . . . . (18) 
cos. 2 A = C0S.2 A — sin.2 A, or cos. 2 A = 2 cos.^ A — 1, 
or cos. 2 A=l— 2sin.2 A .... (19); 
and from the last two of these we immediately get 

cos. A = V 1 -f 1 cos. 2 A, sin. A = V h— h cos. 2 A (20); 

and, therefore, by division, 

I 1 — COS. 2 A ^ 11-1- COS. 2 A 

tan. A = ,-— — -, cot. A = -— ^ ^^^-/T • • • • (3i)j 

^1 1 -|- COS. 2 A' n] 1 — COS. 2 A ^ ^' 

from which we get two new expressions for cos. 2 A, viz. 
_ . 1 — tan.2 A cot.^- A — 1 .„_. 

^°"'^=T+T^Z^ = cot.^A+l • • • • ^''> 
If instead of A we write 45° — A, then since cos. (90° — 3 A) =-' 
sin. 2 A, we have 

. £),_! — tan.2 (45° — A) _ cot.^ (45° — A) — 1 
^^^' " ~ 1-f tan.2(45° — A) ~ cot^ (45°-^y+l' 
It maybe worth while to remark that the radical in the above expres- 
sions for tan. A, cot. A, may be removed by multiplying the numerator 
and denominator of each fraction by its numerator: we thus have 
^ 1 — COS. 2 A ^ 1 + COS. 2 A 

tan. A=: : — -— r — , cot. A — — ——Tnr — • 

sm. 2 A ' sm. 2 A 



38 PLANE TRIGONOMETRY. 

For the tangent and cotangent of a double arc we have, by division, 
. ,,^. sin. 2 A 2 sin. A cos. A , . ,. .,. 

(18), (19), pr— r = -^ . „ ^ : that is, dividing numerator 

^ ^' ^ -" cos. 2 A C0S.2A— sin.2 A' 

and denominator of the second member by cos.sA, or by sin.sA, and 

recollecting that — = tan., and that = cot., we have 

® cos, ' tan. 

tan. 2 A — j _ ^^^2 _^ ""cot.s A— 1 "~ cot.A —tan. A ! .^ox. 

^^ I— tan.^A cot.2A-l ,, ,, , ^^ \ ^^' 

cot. 2A = -^r- T- =-^r — —J— = Kcot.A — tan.A); 

2 tan. A 2 cot. A -* 

which expressions also immediately come from the values of tan. (A 

-{- B), cot. (A 4- B), at (26), by putting A = B. Comparing the above 

value of tan. 2A with the expression (8), art. (26), we have, 2 tan. 2 A 

= tan. (45° + ■^) — tan. (45° — A) ; or which is the same thing, 

2 tan. A = tan. (45° + J A) — tan. (45° — ^ A) . . . . (V). 

Formulas for the secants and cosecants of double arcs are easily 

deduced from those for the cosine and sine, because 

sec.= , and co-ec. = — -. — , thus, from equation (22) above, 

COS. sm, ^ 

« A l-|-tan.2A sec.^A , . . ,,„. 

we have sec. 2 A = i_tan.^ ^ "^ 2-sec.^ A ' ^^ ' equation (18), 

cosec. 2 A = r-^ 7 ■ — r- = i sec. A cosec. A. 

2 sm. A COS. A 

(31.) Another useful class of formulas are those for half arcs; they 
may be easily deduced from the expressions for the double arcs; thus 
putting J A for A, we have from (20). 

sin. § A = V * — i cos. A, cos. i A = Vh -]- i cos. A .... (24) ; 
, „ .^,, , . , 1 — cos. A 1 — COS. A 

also from (21), tan. i A J - — -, t- = -. — -\ 

^ 1 -{- cos. A sm. A 

I 1 + cos.A 1 + cos. A C . . (25). 

cot. I A = n1 — I _ =3 -4 J ^ 

1 — cos. A sm. A -^ 

Other useful values of sin. J A, and cos. J A, are derivable from the 
equation (18) last article, for when I A is put for A the equation is 
sin. A = 2 sin. i A cos. J A ... . (26), and if this be either added to or 
subtracted from 1 = sin. ^ i A -[- cos. ^ ^ A, 

the second member will become in each case a perfect square, viz. 
1 + sin. A = (sin. | A + cos. i A)^ 
1 — sin. A = (sin. i A — cos. J A)^ ; 

hence ^ ~i~ ^^^- -^ "^^ ^^^- 2 ■^~\~ cos. J A 
^1 — sin. A = sin. J A — cos. i A. 
Let A be less then 90°, then the radical must be taken positive in the 
first, and negative in the second expression ; hence, by addition and 
subtraction, sin. J A = i (VI + sin. A — V 1 — sin A) , 

cos. ^ A == i (V 1 + sin. A 4-vl— sin. A) ^ ^' 
By means of these two expressions the accuracy of a table of sines 
and cosines may be examined ; that is to say, from che calculated values 
sin. A, in the table, we may compute, by these equations, the values of 
sin. k A, and of cos. i A; if these agree with the tabular values, found 
by other means, we may conclude that the tables are correct in the parts 
thus verified. Formulas employed in this manner to put the accuracy 



TRIGONOMETRICAL FOEMTJLAS. 39 

of the tables to the test are csXledi formulas of verification. We have 
given three of these, and marked them with the letter (V). 

(32.) The following formulas involving the half sums and half differ- 
ence of two arcs are of frequent application : substitute |(A -|- B) for A, 
and HA — B) for B, in the equations (3), (4), at art. (26) and we have 

sin. A4-sin. B =r 2 sin. A (A-|-B)cos. i(A — B)^ 

cos. A + cos. B = 2 COS. ^ (A + B) cos. | (A— B) 1 ,£,„x . 

sin. A — sin. B =2cos. KA + B)sin. KA — B) f * * * ^'^'^' 



cos. B — COS. A r:^ 2 sin. § (A -f B) sin. i (A — B) J 
and from these w^e get, by division, 

sin. A + sin. B ^ w a , t.n sin. A — sin. B ,,/« i t>\ /oo\ 

7—^ ^= tan. i (A + B) ; T-=cot. i(A+B).(28) 

cos.A + cos.B ^ ' ^' COS. B. — cos. A ^ ^^ ^ 

sin.A— sin.B , sin. A-fsin. B ^ ■> r ^ -D\rnQ\ 

7— = tan. I (A — B) ; — ^ = cot. | (A— B).(29). 

cos.A+cos.B ^ ^ COS. B — cos. A 

In each of these expressions let A = 90°, and we shall have 
1 4- sin. B = 2 sin. (45° + i B) cos. (45° — | B) = 2 sin. ^ (45° + i B) 
COS. B. = 2 COS. (450 + iB) cos. (45° — JB) = 2cos. ^i B — 1, by eq. 18, 
1 — sin. B = 2 cos. (45° -f J B) sin. (45° — i B) = 2 cos.2 (45° -f- 1 B) 

= 2 sin.2 (45° — | B) 
1 — COS. B = 2 sin. (45° + J B) sin. (45° — A B) z= 2 sin. sj B, by eq. 19, 

l±^^tan.(45°+jB),t:=:^-?- = cot. (45o + xB) 
COS. B COS. B 

= cot. (45° — i B), = tan. (45° — | B) 

l±li2.| = tan. (45o + iB),l+i2!4 = eo.. JB. 
1 — sm. B ' '1 — COS. B 

Again, dividing (28) by (29,) we Jiave 
sin. A -f sin. B _ tan. * ( A + B) sin. A — sin. B _ cot. i (A + B) 

sin. A — sin. B ~ tan. i (A — B)' sin. A + sin. B~cot.|(A — B) ^' 

Lastly, substituting A -f- B for A in (26) last article, we have 

sm. (A --f- B) =: 2 sin. ^ (A -f- B) cos. i (A -f- B) ; and dividing this by 

each of the formulas (27) in succession, there results 

s in. (A-f-B) cos. |(A + B) _ s in. (A + B) _ sin. j (A+B) 

sin. A + sin.B ~ cos. § (A — B) ' cos. A -(- cos. B ~ cos. i (A — B) 

sin. (A 4- B) __ sin, j (A + B) ^ si n. (A+B) _ cos. ^ (A + B ) 

sin. A — sin. B " sin. i (A — B)' cos. B — cos. A ~ sin. § {A — B). 

(33.) We shall conclude this chapter on ihe theory of the trigono- 
metrical lines, with two curious and usefal propositions. 

1. To express the sine and cosine of a real arc by means of imaginary 
exponentials. By the exponential theorem* 

€^ = 1 + a; -j-- — -|- -|- ■ 4" &c. where e represents the base of 

Z Z ' O Z' O' 4: 

the Naperian logarithms, that is, e = 3-7182818, &c. For x substitute 
^ V — Ij and — X '^ — 1 successively, and we have these developments 

• See the "Elementary Essay on the construction of Logarithms," p. 68 ; or Young's 
Algebra, jusi published by Carey, Lea, «fc Go. Philadelphia. 



a. V-l=u\ 



(2)5 



40 PLANE TRIGONOMETRY. 

hence, by addition, 

But by art. (30) the series on the right is the development of cos. Xf 

hence cos. rr= 

By subtracting (2) from (^1) we have 

But by art. (30) the series on the right is the development of sin. x', 

hence, sin. x = ^ . 

2 V_l 

2. To develop sin.^ z, cos.^ x, in terms of the sine and cosine of the mid- 
tiples of X. 

Put cos. X -J- sin. a . V 

cos. X — sin. 
then art. ''29), cos. nx -f sin. nx.V — 1 = u^ 

COS. ^i-T — sin. nx .^ — 1 = v^' 
from which, by addition and multiplication, we get 
^71 _}_ ^a _ 2 COS. nx, unq.n — 1 . . . (3). Add together the equations (1) ; 

there will result cos. x = — {u-\- v)-^ and, therefore, 

cos. '^ X =— {n -\- xY = — (v -{■ uy.; hence, by the binomial theorem, 

cos."a;= — \u"-\-nu''~iv-\ 5^— — -u" ^v^-{-S[/C.l, 

1 , , , , nCn — 1) _^ „ , „ 

or, cos.^a; = -^H-"-[-7iv" ^u-\ ^-^r — - v"- -" u^ -f- Sicl', 

adding these equations together, and dividing by 2, we have 

ZQ^.'^x = -^Y—\u7'^'D'^-\-nuv (%"~2-|~'y" ~^)~f" 

''^V^^ ^' ^-^ (^" - 4+ y- ^) -f &c. \ 
2 

But from (3) u" -}- -z;" =2 cos. nx. uv = 1 

u"—^-]-v"~^=2cos.(n — 2)x u^v^ = l 

-M." — ^ -j- -y" — * 1= 2 cos. {n — 4) a; u^v^ = l 

&c. &c. &c. &c, 

hence, by substitution, the developement of cos." .t becomes 

COS." X = — I cos. nx-\-ncos.(n — 2) a; -|- 

'^^''-^} cos.{n-^)x-^&c.\ . . . (4). 

Again, subtract the second of (1) from the first, and we have 

^ • . "^ — "0 

2 sm. X . V — \z=u — V .•. sm. x = : : 

and, consequently, sm . " .r = - . 

(2V - ly 



TRIGONOMETRICAL FORMULAS. 41 

1, Let n be even, then (Algebra, p. 149,) (w — 'oY —{v — w)" ; hence, 

sin." X = {u — vY, or sin." x = rzr:=: {v — w)"; 

(3^—1)1 (2 V—l)n 

and by adding these equations together after having developed (w — ■»)", 
and (v — uy, we have 

2 sin." X = wn_i_yn — nuv(u'^~'^-\-v^ ~'^)4- 

and making the same substitution as before in virtue of (3), and re- 
collecting that, because n is even, (V — 1)''=: ^f- 1, the upper sign 
having place when n is either of the numbers 2, 6, 10, &c. and the lower 
sign when n is either of the numbers 4, 8, 12, &c. we have for the de- 
velopment of sin."a; 

sin."a; =: ^ -^^cos. nx — n cos. (n — 2)a;-|- 

""^"""•^^ cos-Cti — 4)a; — &c.^ . . . (5). 

2. Let n be odd, then (w — •y)" = ( — 1)" (v — %)" = — (v — u)" ; 
therefore, 

1 / N 1 / X 

sm." X = rzrr^ Qii — i^r, or sm." z = -' (v — uY - 

(2V:3]>' (2V:riyi^ ^ ' 

and developing (u — vp, (v — «)" as before, and taking the sum of these 
equations, we have 

2sin."a;= ———.I'u/^ — -u" — nuvOu/'~^—~v'^''^')-{- 

^l^^?^w2^-2(tA"-4_'y"-4)_&C.^ 

But from the equations (2), u" — -y" = 2 sin. nxV — 1, %" ■?;" = 1 ; 
consequently, since (V — l)"~i = q: 1, the foregoing development 
becomes sin."a: == qp — ^sin. nx — wsin. (n — 2)x-\- 

— ^-- sm. {% —^)x — &C.S', 

the upper sign having place when n — 1 is either''of the numbers 2, 6, 10, 
(&c. and the lower sign having place when n — 1 is either of the num- 
bers, 4, 8, 12, &c. The general term of the first series of numbers is 
im -f- 2, that of the second series Am. 
4* F 



PART n. 

ELEMENTS OF SPHERICAL TRIGONOMETRY. 



CHAPTER L 

ON THE SPHERE. 

(34.) A Sphere is a solid whose surface is every where equally dis- 
tant from a certain point within it, called the centre. It may be gene- 
rated by the revolution of a semicircle about the diameter. 

Any line drawn from the centre to the surface of the sphere is called 
the radius ; and the line through the centre having both its extremities 
in the surface, is the diameter. 

A plane surface, or simply a plane, is that in which if any two points 
whatever be taken, the straight line which joins them shall lie wholly 
in that surface. 

A plane may be drawn through any three points, taken at random in 
space, but not through more than three ; for having joined two of the 
proposed points by a straight line, we may pass a plane through this 
line in any direction, and we may turn it round upon this line till it 
arrives at the other point. Three points, therefore, not in the same 
straight line, fix the position of a plane. 

It follows from this, that the common intersection of two planes must 
be a straight line ; for, if among the points in the intersection there be 
three which are not in the same straight line, the two planes passing 
through them must coincide and form but one. 

A straight line is said to be perpendicular to a plane when it is per- 
pendicular to ever}^ straight line in that plane, dravn through its foot, 
or the point where the perpendicular meets the plane. These definitions 
will sufiice for the purpose of establishing the necessary preliminary 
theorems of spherical Geometry. 

(35.) If a sphere be any hoiv cut by a plane, the section must be a circle. 

Let C be the centre of the sphere, and ADB 
the plane section ; draw Cc perpendicular to this 
plane, and from c draw any line cD in the sec- 
tion and terminating at the surface ; then the 
angle CcD must be a right angle. Join CD, then 
wherever the point D may be, CD will always 
be of the same constant length, being the radius 
of the sphere ; and in consequence of the right 
angk c, cD = VCD^ — Cc^ ; hence CD must 
have the same constant length in whatever direc- 
tion it be draT^Ti, that is, the bounding line ADB is the circumference 
of a circle of which c is the centre. 

The circle is, obviously, the larger, as it is nearer to the centre C of 
the sphere, or as its perpendicular distance Cc is less, because CD being 
constant, cD increases as Cc diminishes, and becomes the greatest pos- 
sible when Cc is 0, that is, when the section passes through the centre 
of the sphere ; hence every circle whose plane passes through the centre 




ON TKE SPHERE. 43 

of the sphere is called a great circle of the sphere, and every other a 
small circle. 

It ]s obvious that the circumference of a great circle may be drawn 
through any two points on the surface of a sphere, because a plane may 
be drawn through these two points and through the centre also, but a 
great circle cannot be drawn through three points on the surface, taken 
at random, because then a plane might be drawn through four points 
taken at random ; a circle of some kind, however, may always be drawn 
through three points on the surface of the sphere, since a plane may be 
drawn rhrough them. 

The line Cc from the centre of the sphere perpendicular to the plane 
of the circle passes, as we have seen, through its centre c ; if this line be 
produced both ways to the surface of the sphere, the opposite points P, 
P', are called the poles of the circle. Thus every circle on the sphere 
has two poles diametrically opposite, the diameter which joins them 
being perpendicular to the plane of the circle. The poles of a small 
circle are unequally distant from its plane, the inequality of distance 
amounting to twice Cc ; but in a great circle this inequality vanishes, 
and the poles are equidistant from the circle. 

As the poles of any circle are at the extremities of a diameter of the 
sphere, an infinite number of great circles may be drawn through 
them ; indeed, every circle passing through them will necessarily be a 
great circle, because the entire diameter joining them must be com- 
prised in every plane drawn through them. The distance of any circle 
from either of its poles^ measured upon any of these infinite number of 
great circles, is constantly the same, that is, the distances or arcs PB, 
PD', PD, PA, &c. are equal, because the constant line Pc is the com- 
mon versed sine of all these arcs to the common radius CP ; hence the 
other distances P'B, P'E, &c. must be equal. Every arc of a great circle 
is thus distant from either pole by a quadrant or 90°. 

(36.) Tv:o great circles ahvays intersect in tioo points at the distance of 
a semicircle from each other, that is, the circumferences bisect each other. 
For as the plane of each circle passes through the centre of the sphere 
their intersection must be a diameter common to both circles, and it is 
at the extremities of this diameter that the circumferences cross each 
other. 

From this we learn that if from any point on the sphere two qua- 
drantal arcs can be drawn to two points in any great circle, the distance 
between the points being less than 180°, then the first point must be the 
pole of this great circle ; for it is necessarily the pole of some great 
circle passing through the proposed points, and as only one great circle 
can pass through two points, which are not 180° apart, the pole must 
belong to the circle in question. 

In spherical trigonometry, the arcs of great circles only are concerned, 
and the angle included between two such arcs, that is to say, a sphe- 
rical angle, is measured in a manner analogous to that in which a plane 
angle is measured. For the measure of a plane angle we take the 
intercepted arc of that circle whose centre is at the vertex, and whose 
radius is some assumed unit: in like manner for the measure of a spher- 
ical angle we take the intercepted arc of that circle whose pole is at 
the vertex, and Avhose radius is some fixed unit, viz. the radius of the 
sphere on whose surface the angle is : thus, in the foregoing figure the 
spherical angle DPD' is measured by the intercepted arc GlGl' of which 
the pole is P, and radius, CGI, that of the sphere. 

It is as easy to justify the propriety of adopting this mode of mea- 
suring spherical angles as it is to justify the method of measuring 
plane angles, for in both cases the intercepted arc varies as the angle j 
this, by the by, is true of the intercepted arc DD' of any small circle 



44 SPHERICAL TRIGONOMETRY. 

whose pole is P, but we are compelled to refer the measure to a great 
circle, in order that all the trigonometrical lines concerned in the 
same inquiry may be related to a common radius, for as v^e have before 
remarked, the sides of a spherical triangle are always arcs of great 
circles. 

From what we have just said it appears that a spherical angle DPD' 
has the same measure as either of the equal plane angles Q.CGL', DcD', 
&c. situated in the planes of the circles whose common pole is P, and 
whose sides are formed by the intersection of these planes with those 
of the two great circles, forming the sides of the spherical angle. If at 
P tangents were drawn to the two great circles PD, PD', and in their 
planes they would obviously include the same angle as the lines CGI, 
CO,', to which they are parallel ; indeed if we conceive the plane of 
the circle HQ.Q.', to move parallel to itself towards the pole, P, the 
path of C being along the line CP, the angle Q.CQ.' will successively 
coincide with GlCQ.', DcD', &c. till C coincides with P, when the 
lines CO., CO,', will become tangents to the circles at P, and will remain 
each in the plane along which it has moved ; hence the measure of the 
angle included between these tangents is also the measure of the spherical 
angle. 

(37.) If in the plane of HQ.I perpendiculars be drawn from C to 
each of the planes of the circles PGIP', PQ,'P', these will be perpendi- 
cular to the lines CO., CGI', and will therefore, include the same angle, 
which angle will be measured by the arc of HGlI, which the said per^ 
pendiculars intercept ; but these perpendiculars will meet the surface 
at the poles of the circles to whose planes they are perpendicular ; hence 
the great circle distance beticeen the poles of two intersecting great circles 
meo.sures their angle of intersection. 

Every great circle which passes through the poles of another is at right 
angles to it. Thus the great circle PDQ.P', through the poles of 
HQ.Gl'1, is at right angles to HGlGl'I; for if a tangent were dra-UTi to 
PQ.P' at the point Q. it would be in the same plane with and parallel to 
CP, and if a tangent were drawn to HQ.I at the point Q, it would be in 
the same plane with and parallel to CH ; hence if these two tangents 
were to move simultaneous^ to themselves, the path of their point of 
concourse Q, being along GIC, they would necessarily coincide with the 
perpendiculars CP, CH, when Q, arrived at C : these tangents, there- 
fore, form a right angle ; hence the great circles are perpendiculars to 
each other, or the spherical angle at Gl is a right angle. 

(38.) Any one side of a spherical triangle is less than the sum of the 
other tv:o. 

Let ABC be any spherical triangle, and O 
the centre of the sphere ; draw the radii OA, 
OB, OC, then there will be about O three 
angles in three distinct planes respectively, 
measured by the arcs AB, BC, CA. Let AB 
be the greatest of these arcs, then it will only 
be necessary to show that AB < AC -j- CB, or 
that AOB < AOC + BOC. In the plane of 
AOB draw any line A'B', and then draw OD, 
making an angle B'OD equal to BOC ; make 
OC equalto OD, and join CB', CA' 

Then since by construction the two sides B'O, OD, and the included 
angle, are respectively equal to the two sides B'O, OC, and the included 
angle, B'D = B'C. But in the plane triangle A'B'C. A'B'^ A'C -f 
B'C .-. A'D<A'C; hence the two sides OA', OD, of the triangle A'OD, 
are equal to the two sides OA', OC, of the triangle A'OC',but the third 
side A'D of the former is less than the third side A'C of the latter, and, 




ON THE SPHERE. 



45 





consequently, A'OD < A'OC ; hence, since B'OD has been made equal 
10 B'OC, it follows that 

A'OD + B'OD = A'OB' < A'OC + B'OC /. AB < AC + CB. 
(39.) The sum of all the three sides of a sfherical triangle is less than 
the circumference of a great circle. 

Let ABC be any spherical triangle ; produce the 
sides AB, AC, till they meet again in D, then the 
arcs ABD, ACD, will be semi-circumferences, since 
(36,) two great circles always bisect each other. 
But in the triangle BCD we have BC < BD + CD, 
and, consequently, by adding AB -|- AC to both, we 
shall have AB + AC + BC < ABD + ACD; 
that is to say, the sum of the three sides is less than 
a whole circumference. 

By help of this theorem we may show that the sum of the sides of any 
spherical polygon whatever is less than the circumference of a greett 
circle. 

Take the spherical pentagon ABCDE for ex- 
ample. Produce the sides AB, DC, till they 
meet in F ; then since BC < BP 4- CF, the peri- 
meter of the pentagon will be less' than the quad- 
rilateral AEDF. Again, produce the sides DE, 
B A, till they meet in G ; we shall have EA < 
EG -f- AG ; hence the perimeter of the quadri- 
lateral AEDF is less than that of the triangle 
DFG ; which last is itself less than the circum- 
ference of a great circle ; the perimeter of the original polygon is, there^ 
fore, less still. 

(40.) If from the three vertices of a spherical triangle, taken as poles^ 
arcs be described, forming a new triangle, then the vertices of the new tri- 
angle loill be the poles of the other triangle. 

For let ABC be any spherical triangle, D 

and with the pole A, and circular radius ^i'^^lS^ 

AG equal to a quadrant, describe the arc, /^^^^ PC^\ 

EF ; in like manner with the pole B and X /\ \ 

same radius describe the arc FD, meeting / / \ \ 

the former in F; and, lastly, with the pole -mf I \ \^ 

C and same radius describe the arc ED, I-",.,. £___—- ^''c '" \ 

completing the spherical triangle DEF. / B/ \ \ 

Then, because the arcs, whose poles are / I \^^ 

A and C, intersect at E, the points A, C, are 33~"" — J — Ij 

each 90° distant from E ; and as the arc AC ^ 

is less than 180°, E must be the pole of AC (36). In like manner is it 

shoM^n that F is the pole of AB, and D the pole of BC. 

The triangle DEF is sometimes, from the mode of its construction, 
called the polar triangle, and the original one ABC the primitive tri- 
angle. 

(41.) Any angle of the primitive triangle is the supplement of the side 
opposite to it of the polar triangle, and any angle of the polar triangle is 
the supplement of the side opposite to it in the primitive triangle. 

For EH being the radius of HL is = 90°, and FG being the radius of 
GK is also = 90°, and the sum of these radii, namely, EF -j- GH = 180°, 
therefore, GH, which is the measure of the angle A, is the supplement 
of the side EF opposite to it. In like manner it is shown that B is the 
supplement of DP, and C the supplement of DE. Again, BI being the 
radius of ID, and CM the radius of MD, the sum of these MI -f- BC 
=: 180° • therefore, BC is the supplement of MI, which measures the 



46 



SPHEEICAL TRIGONOMETRY. 




angle D. On account of the property just demonstrated, the triangles 
ABC, DEF, are frequently called supplemental triangles. 

It is proper to remark here, as Legendre has done, that besides the 
triangle DEF three others might be formed by the intersection of the 
three" arcs DE. EF, DF. But the proposition immediately before us 
is applicable only to the central triangle, which is distinguished from 
the others by the circumstance that the tTTo angles 
A and D (see preceding fig.) be on the same 
side of BC, the t\ro B and E on the same side 
of AC. and the t\ro C and F on the same side of 
AB. 

(42.) From the foregoing proposition it follows 
that three angles of every spherical triangle are to- 
gether greoMr than tico right angles, and less than 
six. 

For the sides of the supplemental triangle DEF are together less than 
four' right angles (39). and as these are supplements of the angles A, B, 
C. and therefore when added to them make six right angles, these last 
must together exceed two right angles. But they cannot amount to six 
right angles, for in that case the sum of the sides of the supplemental 
triangle would be 0. which is absurd. Hence, unlike plane triangles, 
a spherical triangle may have all its angles right angles or all obtuse 
angles. 

(43.) The foregoing geometrical properties comprise all that we 
require, for the foundation of the analytical theory of spherical Trig- 
onometry: we need not. therefore, enumerate any more. We shall, 
however, in conclusion, endeavour to establish the fact that the arc of a 
great circle joining tiro points is the shortest line that canbe draicn an the 
sphere from the one to the other. 

The" following proof of this property is by Legendre. 

Let AIS'B be the arc of the great circle which joins the points A and 
B : and without this line, if possible, let M be a point in the shortest 
path, between A and B. Throusfh the point M iraw 
MA. MB. arcs of great circlesj and take BX = MB. -^ 

Then, by (3S). the arc A]SB is shorter than AM -f 
MB ; takeBX =BM, respectively from both: there will 
reniain AIXk^ AM. 

]S"ow, the distance of B from M, whether it be the same / 

with the arc B]M or with any other line, is equal to the / 

distance of B from X : for, by making the plane of the / 

great circle B^I revolve about the diameter, which ^/ 
passes through B. the point M may be brought into the \ 
position of t£e poitit ^S" ; and the shortest line between M \ 

and B. whatever it mav be. will then be identical -uith that \ ^ 



B 



between ZS" and B : hence the two paths from A to B, one 
passing through M, the other through X. have an equal 
part in each, the part from M to B equal to the part from N to B. The 
first path is the shorter by hypothesis : hence the distance from A to M 
must be shorter than the distance from A to X; which is absurd, the 
arc AM being proved greater than AX : hence no point of the shortest 
line from A to B can be out of the arc ANB : hence this arc is itself 
the shortest distance between its two extremities. 



ON THE SPHERE. 4^ 

CHAPTER II. 

INVESTIGATION OF FORMULAS, AND RULES FOR THE SOLUTION OP 
SPHERICAL TRIANGLES. 

(44.) Let ABC be a triangle traced 
on the surface of a sphere of which 
the centre is O, and the radius equal 
to the linear unit. The angles of this 
triangle we shall represent by the 
letters at their vertices, A, B, C, 
and the sides opposite to them by the 
small letters a, b, c\ so that having 
drawn the two tangents AD, AE, to meet the radi OB, OC, produced 
through the other extremities of the arcs AB, AC, we shall have 

AD=tan.c = ?^^,AE=:tan.i '^'^ 




OD = sec. c = , OE = sec. b = ^ 



COS. y 

(1). 



COS. b 

Draw DE, then m the two triangles ODE, ADE, we have (17^ 
DE2 = 0E2 -}- 0D2 — 2 OE-OD cos. a 
DE2 = AE2-f AD2 - 2 AE'AD cos. A ; 
recollecting that (p. 43-44) the plane angle DAE measures the spheri- 
cal angle A. Substituting in these equations the values given by (I), 

they become DE^ = sec. ^b -j- sec. ^c '- 

COS. b cos. c 

T^„2 * pz. I . 9 2 sin. ^ sin. c COS. A , , 

DE2 = tan. ^b-{- tan. ^c— — .-, by subtraction 

COS. b COS. c 

2 

Q= 1 -f- 1 + (sin. b sin. c cos. A — cos. a) 

cos. b COS. c 

Tx I • 1 • 1. COS. b COS. c , . , 

Hence multiplying by , and transposing, we have 

cos. a — COS. h COS. c -j- sin. b sin. c cos A ; which is a general expres- 
sion for the cosine of any side in terms of the other two sides, and their 
included angle. If we had taken the side b instead of a, the other two 
would have been «, c, and their included angle B ; and if we had taken 
the side c the other two would have been «, b, and their included angle 
C ; we have, therefore, the three following symmetrical equations, viz. 
COS. a = COS. b COS. c -|- sin. b sin. c cos. K\ 
cos. b = COS. a COS. c -|- sin. a sin. c cos. B )» . . . , (A) ; 
COS. c = COS. a COS. b -f-sin. a sin. b cos. C J 
and these equations embody the whole theory of spherical trigonometry 
and are sufficient to supply rules for the solution of every case. 

(45.) Some interesting geometrical properties flow also from these 
equations. 

1. Suppose two sides b, c, of the triangle are equal, that is, let it be 
isosceles, then it will follow from the two last of these equations that, 
like as in the isosceles plane triangle, the migles opposite the equal sides 
will be equal. For taking the difference of these two equations on the 
supposition that b =c; we have = sin. a sin. b cos. B 

— sin. a sin. b cos. C ; and, consequently, B = C. 

2. If a = b = c, then it is in a similar manner proved that A = B = 
C, that is, every equilateral spherical triangle is equiangular. 

3. The arc which bisects the vertical angle A. of a spherical isosceles 



48 SPHERICAL TRIGONOMETRY. 

triangle also bisects the base a. For let p represent this bisecting arc, 
and m, m' the parts into which it divides the base, then the two spherical 
triangles thus formed give, by the above equations, 

cos. m = cos. b cos. p -\- sin. b sin. p cos. J A 
cos. m' = cos. a cos. p -j- sin. a sin.p cos. | A ; 
therefore, since by h}^othesis a = b, we have m = m', that is, the arc 
bisecting the vertical angle also bisects the base, and the student will 
find no difficulty in further showing that this same arc is also perpen- 
dicular to the base. 

4. If tic sides and the iticluded angle in one triangle are eqical to 
two sides, and the included angle in another, the thirds side of the one 
must be equoZ to the thirds side of the other. This is obvious from the fijst 
of (A\ which shows that cos. a, and therefore a, becomes fixed when 
the other two sides b, c, and their included A, is fixed ; moreover, the 
rernaining angles of the one triangle o.re equal to the remaining angles of 
the other i for by the second and third of (A), cos. B, cos. C, and there- 
fore, B, C, become fixed Avhen a, b, and c, are fixed. 

5. If the three sides of one triangle are severalhj equal to the three 
sides of another, the three angles of the one are also severally equal to. 
those of the other, the equal angles being opposite to the equal sides. 
For with fixed values for a, b, c, the formulas (A) give fixed values for 
cos. A, cos. B, COS. C, and, therefore, for A, B, C. We may, in like 
manner, infer the equality of the sides from that of the angles, but per- 
haps the inference is a little more obvious from the equations (B), p. 
51, following. 

In these deductions the student will observe that we have abstained 
from saying that the triangles are equal in all respects as in the analogous 
theorems of plane geometry; because two spherical triangles may exist, 
of which the several parts of the one may be equal to the several parts 
of the other, and yet not admit of coincidence, as plane triangles would 
under like conditions. Thus, if two plane triangles, of which the sides 
in the one are equal to those in the other, be joined together by a cor- 
responding side of each, and if we turn one of the triangles about this 
common side either above or below the plane on which they are situated 
till it comes to that plane again, v/e know that we shall thus obtain a 
perfect coincidence between the two ; but if the sides of the triangles 
thus joined are the chords of two spherical triangles, these triangles will^ 
as we have seen, have all their parts equal, each to each, because, the 
chords being equal, the arcs must be equal, and yet it is very plain that 
the corresponding parts of the two triangles cannot be brought into 
coincidence as in plane triangles, and only in the particular case in 
which the two triangles are isosceles can they coincide, by being laid 
the one over the other. We cannot therefore, say, as in plane triangles, 
that two triangles, whose corresponding parts are equal, have equal sur- 
faces, without distinct proof. This proof will be given in Part iv. 

We shall add here but one more inference from the fundamental 
equations (A). 

6. By the first of (A) if the sides b, c, are fixed, cos. a will necessarily 
diminish as cos. A diminishes ; that is, a will increase as A increases: 
hence if tv:o triangles have tioo sides in the one equal to two sides in the 
other, but the included OMgle in the first greater than the included angle 
in the second, then the third side of the first triangle must be greater than, 
the third side of the second. 

Let us now proceed with the analytical discussion. 

The three general equations above involve all the six parts of a tri- 
angle, the sides, and the angles ; and in order to solve them, fewer than 
three of these parts will be insufficient ; but, knowing any three, the 
others may be determined from them by the usual algebraical process 



ON THE SPHERE. -IW 

of elimination; yet, as in the general formulas for the solution of plane 
triangles, so here, the result thus obtained would require considerable 
modification in certain cases to fit them for logarithmic computation, 
and on this account it is better to deduce particular formulas by a less 
direct process. Thus, ia order to ascertain the relation between the 
sides and opposite angles of a spherical triangle, we proceed as follows, 

, .^ . „ , ■ ^AN A cos. « — COS. 6 cos. c ,,^ 

(46.) From the equation (A), cos. A = -. — -— : ... (1) 

sm. b sm. c 

V sin. " b sin. " c — (cos. a — cos. b cos. c)^ 



.'.sin. A = VI — COS.- A— , . 

sm. b sm. c 
or, since sin. ^ b sin. " c — {l — cos. 2 b) (1 — cos. 2 c) 

sin. 



_ VI — COS. 2 a — cos. 2 6 — cos. 2 c -(- 2 COS. a cos. b cos. c ^ ri^ 



sin. b sin. c 



sin. A_ VI — cos. ^a — cos. 26 — cos. 2 c -]- 2 cos. a cos. b cos. c 
sin. a sin. a sin. b sin. c 

Now the second side of this equation is plainly of such a form, thatj 

however we interchange the quantities a, b, c, the value of the express 

sion remains imaltered ; so that if we had set out with cos. B, as given 

by the second of (A), instead of with cos. A, we should have had the 

, ^ sin B , sin. A sin. B sin. C 

very same result lor-^ — r\ hence —. = —. — r= —■ .... (3). 

svd.b sm.a sm. sm. c 

that is, in any spherical triangle the sines of the sides are to each other as 
the sines of the opposite angles ; so that when two of the three given 
quantities are a side and its opposite angle, the unknown, which is 
opposite to the third given quantity, may be determined by a simple 
proportion, or by an easy logarithmic process. 

(47.) The equation (2) above might serve to find an angle, from 
knowing the three sides j it is, however, much less simple than the 
original expression (1), but neither of them are adapted to logarithms. 

In order to obtain one that is adapted, add 1 to each member of (1) 
and there results (form 24, p. 38), 1 + cos. A = 2 cos. 21 A 
_ COS. a -\- sin. b sin, c — cos, b cos.c _ c os, a — co s . {b-{- c) _ 
~ sin. b sin. c ~ sin. b sin. c ' 

but a and b -\-c are respectively the difierence and sum of the two arcs 
i (a -{- b-\- c), and k {b -\-c — a); hence (form 4, p. 32), 
cos. a — COS. (b-{- c) = 2 sin. i {a -[- b -\- c) sin. i{b-{-c — a); 
therefore, putting S for the sum of the three sides, we have 
eos.jA= |£i?jSsin.(iS-«) _ _ 
N sm. b sm. c 
If instead of adding 1 to each side of (1) art. 46 we subtract each side 
from 1, and proceed as above, we shall obtain for sin. J A the value, 
sin.iA= I sin.(iS-6)sm.aS_--.) _ 

^J sm. b sm. c 

and, by dividmg this equation by the former, we have 

tan. |A= 1 sin, (j S^T)~sin. ( ^ S — c) . . . . ^'^■^ 
n1 sin. h S sin. (J S — «) _ 
and all these expressions are adapted to logarithms. 

It is unnecessary to put down the corresponding expressions for the 
other angles, as they may be obtained from these by simply changing 
the letters : thus for sin. I- B, we have, by changing A for B and b for a 

, ,^N 1 . •■ • -r. 1 sin. (JS— a)sin.(iS — c) 

in (2), the formula sm. | B = ■ — — ~ ^-, 

^ ■' "J sm. a sm. c 

5 G 



50 SPHERICAL TEIGONOMETRY. 



, sm. i A sm. a sm. (^ S — h) 

■whence - — ——-— - — -- — y-- { ; 

sm. i B *J sm. h sm. (§ S — a) 
from which it appears that if a > ^, sin. \ A> sin. \ B, and therefore 
A> B ; also if ^^ > a, sin. i B > sin. \ A; and therefore B > A. 
Consequently the greater side is always opposite to the greater angle, 

. . sin. J a 
If =c, the equation (2) becomes sm. J A =—t- — r- ■ 

(48.) We have thus got convenient formulas for the determination 
of the unknown parts, when two sides and an opposite angle are given, 
when two angles and an opposite side are given, and when all the three 
sides are given. We shall now seek the solution to the case in which 
two sides and the included angle are given, or two angles and the in- 
terjacent side ; that is to say, we shall proceed to deduce an equation 
involving only the four quantities a, b, A and C. 

For COS. c in the first of equations (A) substitute its value, as given 

by the third, and there results, after putting 1 — sin. -b, for its equal 

COS. ^Z*, COS. a = cos. a — cos. a sin. ^b-\- sin. a sin. b cos. b cos. C -|- 

sin. b sin. c cos A ; or cancelling cos. a on each side, dividing by sin. &, 

and transposing, cos. a sin. b = sin. a cos. b cos. C -j- sin c cos. A . (1). 

For sin. c in this equation substitute its value given by (3, p. 49), viz. 

sin. « sin. C , . , • z. • i i-« 

sm. c = — : i : and it becomes cos. a sm- o = sm. a cos. o cos. O 

sm. A 

, sin. a sin. C COS. A , .,..,. 
-[ : ^ that IS dividing by sm. a, 

cot. a sin. b = cos. b cos. C + sin. C cot. A ; 
which is the equation we proposed to deduce, and from which we at 
once get an expression for cot. A, when the two sides a, b, and their 
included angle C, are given, or for cot. a when the two angles A, C, and 
interjacent side b are given. The remaining parts of the triangle may, 
obviously, be found by the relation (p. 49) between the sides and oppo- 
site angles ; but if the third side, in terms of the other two, and the 
included angle, is required in a single formula, we must then recur to 
the fundamental equations (A), which obviously furnish that formula. 
But neither this nor that which we have just deduced are calculable 
by a single logarithmic operation ; by the introduction, however, of a 
subsidiary arc the solution may be conducted by logarithms, although 
two operations will be necessary. But we shall explain this artifice 
in the next chapter, which will contain the practical application of the 
formulas deduced in this. 

(49.) It now only remains for us to furnish a formula for the side of 
a spherical triangle in terms of the three angles, and this we may 
easily do by help of the formulas already given for an angle in terms 
of the sides, availing ourselves of the property of the supplemental tri- 
angle, viz. that the angles and sides of this are supplements of the sides 
and angles of the former (41). For let the formulas (47) refer to the sup- 
plemental triangle of that in question, then, by marking the letters of 
the former with an accent for distinction sake we have A = 180" — «, 

a' = 180=^ — A,b' = 180° — B, c' = 180° — C, S' = 540° — S; 
S' being the sum of the sides of the triangle in (47), and Sthe sum of the 
angles of the triangle with which we are now occupied. Consequently, 
cos. I A' =r cos. (90° —ia)= sin. i a, sin. b' = sin. (180° — B) == sin. B 
sin. c' = sin. (180° — C) = sin. C, sin. J S' = sin. (270° — | S) = 
— cos. i S ; sin. (J S'—a') = sin. [90° — (J S— A)] = cos. (J S — A); 
therefore by substituting these values the formula (2) becomes 

. , 1 —COS. JS COS. (i S— A) 

sm. ha= ^ ■ - -P ■ ^ ,• 

sm. B sm. C 



INVESTIGATION OF FORMULAS. 51 



^ ^ ^ ^ . , I COS. (^S— B)cos.(^S — C) 

and the other two become cos. ^a — ^ ^ , — -^--^- ■' 

sm. B sin. C 

1 _ [ — cos . jScos. ( jS — A) 
tan. «^->lcos.aS-B)cos':(rS = C)- 
As ^ S exceeds 90° but falls short of 270° art. (42), cos. J S is always 
negative, and, therefore, the numerators, of the first and third of these 
expressions although appearing with a negative sign, are in reality- 
positive. 

(50.) By means of the polar triangle it is obvious that we may, in all 
cases as well as in this, convert any formula involving the sides and 
angles of a triangle into another, similarly involving the angles and 
sides ; the sides in the one formula being replaced by the angles oppo- 
site to them in the other, and the angles being replaced by the opposite 
sides. To effect this change we need only write, instead of sm. and cos 
in the original formula, sin. and — cos. of the opposite arc, whether side 
or angle. 

Thus the fundamental equations (A) become in this manner changed 
into the following 

COS. A -— COS. a sin, B sin. C — cos. B cos. C 1 
cos. B = cos. b sin. A sin. C — cos. A cos. C > . . . (B). 
COS. C — cos. c sin. A sin, B — cos. A cos. B ) 
which plainly show that if the three angles of one triangle are equal to 
the three angles of another^ the sides of the former must also be equal to 
those of the latter ; and also that if two angles B, C, and interjacent side, 
a, of one triangle are respectivelij equal to two angles, and the interjacent 
side of another, the remaining angle A of the one miist be equal to the re- 
Toaining angle of the other ; and thus all parts of the one triangle are 
equal severally to those of the other. 

(51.) The theory now delivered is sufficient for the solution of every 
case of spherical triangles ; but we shall add two more theorems appli- 
cable to the case in which the two sides and included angles are given 
to find the other angles, and to that in which two angles and the inter- 
jacent side are given to find the other sides. These theorems have the 
advantage of being very simple, and are of a form easily retained in 
the memory. They were first given by Lord Napier, and are known 
by the name of Napiefs Analogies. 

By the equation (1), page 50, we have 

sin. c COS. A = cos. a sin. b — sin. a, cos, b cos, C, 

Similarly, 

sin, c cos, B = cos. b sin. a — sin. b cos. a cos, G 

.'. sin, c (cos. A + cos, B) — sin. {a-\-b){\— cos. C) . . . . (1). 
Now from the equations (3), page 49, we have 

sin. A sin. c = sin. a sin. C 

sia. B sin. c = sin, b sin. C 



.*. (sin, A ± sin. B) sin. c = (sin. a ± sin. b) sin. C . . (3). 
Dividing (2) by (1) there results 

sin, A ± sin. B _ sin. a i sin. b sin. C 

cos. A -f- COS. B "" sin. {ci,-\-b) ' \ — cos. C 
that is, arts. (32) and (31) taking the upper and lower signs separately. 
/ . ■ ^. COS. h{a — b) , ^ 

tan. 1 (A + B = -^, — ~ cot. h C 

' COS. ^{a-\-b) 

^ sin. I {a — b) „ 

tan. i (A — B) = ~^~r-r^ cot. A C. 
sm. I {a -\- b) 
For the supplemental triangle the corresponding formulas are 
, , ... cos § (A — B) ^ 

tan. iCa 4-b) — Vic — i—^ tan. i c - 

^^ ~ ' cos. ^ (A 4- B) 



52 SPHERICAL TRIGONOMETRY. 

tan. i (a — b) = ^' , ~ J, tan. i c : and these are the four 

" ^ ' sin. i (A -|- B) 

equations which furnish the Analogies of Napier, viz. 
COS. |(a + 5) : COS. kia — b) • ■ cot. * C : tan J (A + B) \ .ox 

sin. i (a + *) : sin. H« — ^) : : cot. i C : tan. HA — B) 5 ' * ^*>' 
COS. i (A -f B) : COS. l (A — B) : : tan. i c .: tan. I {a -|- h) \ /o-. 

sin. i (A + B) : sin. K(A — B) : : tan. J c : tan. \{a — b)S ' ' ^ >'^ 
As the arcs I {a — b), and | C, are always less than 90°, the two 
means in the first of these analogies are positive, and, therefore, the 
two extremes must have the same signs, that is, they must either be 
both positive or both negative : hence i {a -\- b), and J (A -]- B), must 
either be both acute or both obtuse, and consequently the arcs a -\-b, 
A + B, must be either both less or both greater than 180°. From this 
circumstance we may always avoid doubtful solutions to the cases in 
which the given parts are two sides and an opposite angle, or two 
angles and an opposite side, as will be exemplified ia next chapter. 



CHAPTER III. 

SOLUTIONS OF THE DIFFERENT CASES OF SPHERICAL TRIANGLES. 

(52.) We are now to show the application of the preceding theory to 
the actual determination of any of the six parts of a spherical triangle 
when three of them are known ; and as in Plane Trigonometry, so here, 
we shall find it convenient to begin with right-angled triangles. 

Right-Angled Spherical Triangles. 
The formulas for which all the rules for right-angled triangles are 
deri-v^ed are those marked (A), (B), and 3, (p. 49), in the preceding 
sin. A sin. B sin. C 

chapter, VIZ. — =—. — — = - ... (1) 

•^ sm. a sin. b sm. c ^ 

COS. A = COS. a sin. B sin. C — cos. B cos. C ) 
cos. B = cos. b sin. A sin. C — cos. A cos. C [■ ... (2) 
cos. C = cos. c sin. A sin. B — cos, A cos. B } 
COS. a = cos. b cos. c -j- sin. b sin. c cos. A ^ 
cos. b = COS. a COS. c -{- sin. a sin. c cos. B > ... (3). 
COS. c = COS. a COS. b -\- sin. a sin. b cos. C j 
Let ABC be a spherical triangle, right-angled -rj 

at C ; then from the first of these formulas we 
have, since sin. C = 1, the equations 
sin. a = sin. c sin. A, sin. b = sin. c siii: B . . (4). 

Two difierent expressions for sin. a, sin. b, may 
also be obtained from the first and second of (2). "^ 
Thus C being 90° these two equations give cos. A = cos. a sin. B, 
cos. B = COS. b sin. A . (5) ; substituting in these the values of sin. A, 
sin. B, as deduced from (1) they become 

. cos. a . . . , .„ cos. 5 . - . 

COS. A = sm. A sm. b, cos; B = - — - sm. B sm. a 

sm. a ' sm. b 

.*. sin. b = tan; a cot; A, sin. a = tan. b cot. B. 
For the hypotenuse c we get from the third of (2) the expression 
COS. A cos. B . .„ 

cos. c = -: — -, — — = cot. A cot. B, 

sm. A sm. B ' 

and from the third of (3) the expression cos. c = cos. a cos. b . . (6). 
In the equations (5) substitute for sin. A, sin. B, their values in (4) 
and for cos. a, cos. b, their values in (6), and they then take the form 




RIGHT-ANGLED TRIANGLES. S8 

COS. A = tan. b cot. c, cos. B = tan. a cot. c . . . (7). 
Collecting together all these equations, we have 

sin. a — tan. h cot. B = sin. c sin. A 

sin. b — tan. a cot. A = sin. c sin. B 

cos. c = cot. A cot. B = cos. a cos. b 

cos. A = tan. b cot. c = cos. a sin. B 

COS. B = tan. a cot . c = cos. b sin. A ; 
and these furnish solutions to every possible case of right-angled tri- 
angles ; for it is plain that whichever two of the five quantities a, b, c, 
A, B, are given, any one of the others may be immediately found by 
one or other of these equations. Instead, however, of deducing from 
these five equations so many distinct rules for the solution of the various 
cases, the whole, by help of an ingenious contrivance, may be compre- 
hended in two rules of very remarkable simplicity. 

Before announcing these rules we shall, however, just stop to men- 
tion an inference from the first of this group of equations which will be 
useful hereafter, viz. that/rom any point on a sphere to a given great cir- 
cle the shortest great circle arc that can be drawn is the perpendicular; for 
by the equation referred to sin. a exceeds sin. c, since sin. A is less than 
1. If the point is the pole of the proposed great circle, then, indeed, 
(p. 43) sin. a = sin. c, and sin. A = 1,_ all great circle arcs from the 
point to the circle being perpendicular. From the last of the preceding 
equations we infer that cos. B, cos. b, always have the same sign, that is, 
either side is of the same affection as its opposite angle. From the mid- 
dle equation we see that the hypotenuse is acute if the sides are of the 
same afiection, or if the angles opposite to them are of the same affection, 
but otherwise the hypotenuse is obtuse. 

The rules to which we have adverted above were invented by Baron 
Napier, the celebrated inventor of logarithms, and are called Napier^s 
Rules for the Circular Parts. We shall now explain them. 

In a right-angled triangle we are to recognise but five parts, viz. the 
three sides and the two angles A and B. If we take any one of these as 
a middle part, the two which lie next to it, one on each side will be ad- 
jacent parts : thus taking A for a middle part (last fig.), b and c will be 
the adjacent parts ; if we take c for the middle part, A and B will be the 
adjacent parts ; if we take B for the middle part, c and a will be the 
adjacent parts; but if we take a for the middle part, then, as the part 
C is not recognised we do not consider it as intervening between a and 
b, and, therefore, we call in this case B and b, the adjacent parts ; and, 
lastly, if b is the middle part then the adjacent parts are A and a. The 
two parts immediately beyond the adjacent parts, one on each side, still 
disregarding the right-angle, are called the opposite parts ; thus if A is 
the middle part the opposite parts are «, next to the adjacent part b, and 
B next to the adjacent part c. This being understood, Napier's two rules 
may be expressed as follows, carefully observing to use the complements 
of the two angles and of the intervening hypotenuse instead of these pojrts 
themselves. 

I. Rad. X sin. middle part = product of tan. adjacent parts. 

II. Rad. X sin. middle part =. product of cos. opposite parts. 
Both these rules may be comprehended in a single expression, thus 

rad. sm. mid. =prod. tan. adja. = prod. cos. opp.; 
and to retain this in the memory we have only to remember that the 
vowels in the contractions mid., adja., opp., are the same as those in the 
contractions sin., tan., cos., to which they are joined. 

That these rules comprehend all the equations given above will be 
seen by taking a, b, c, &c. in succession for the middle part, as in the 
subjoined table, keeping in mind the condition just stated, that instead 
of A, B. and c, we are to use their complements. 
5* 



54 SPHERICAL TRIGONOMETRY. 




As in the solution of right-angled triangles two parts are given to 
find a third, vre mast ia the application of uN'apier's rule choose for the 
middle of these three parts that Tvhich causes the other two to become 
either adjacent parts or opposite parts. 

EXAMPLES. 

(53.) 1. In the right-angled triangle ABC are given the two perpen- 
dicular sides, viz. a = 48=^ 24' 16", b = 59° 38' 27", to find the hypote- 
nuse c. 

Here the hypotenuse being made the middle part the other two will, 
obviously, be the opposite parts, being separated from the hypotenuse 
by the intervening angles A, B. Hence by the rule 
rad X sin. comp. c = cos. a X cos. b ; 
, cos. a cos. b 

that IS, rad. cos. c — cos. a cos. b .". cos. c = : and as cos. a, 

rad. ' 

cos. b, are both positive, cos. c is positive, and, therefore, c is acute, 
rad. . . . 100000000 

cos. a 48° 24' 16" . 9-8220819 

cos. b 59 38 27 . . 9-7036515 



cos. c 70 23 42 . 9-5257334. 

2. In the spherical triangle ABC. right-angled at C, are given b = 
46° 18' 23", A = 34= 27' 39", to find the other oblique angle B. 

Making B the middle part, the other two will be the opposite parts. 
Consequently, by the rule, rad. X sin. comp. B = cos. b X cos. comp. A j 

J T. J, • A -n COS. b sin. A 

that is, rad. cos. B = cos. b sm. A .-. cos. B = ■ ; 

' rad. ' 

and as cos b, sin. A. are both positive, B is acute, 

rad.' .... 10-0000000 

COS. b 46° 18' 23" 98393535 

sin. A 34 27 39 9-7526957 



COS. B 66 59 25 9-59-20492. 

3. In the spherical triangle, right-angled at C, are given the two per 
pendicular sides, viz. a = 116= 30' 43", b = 29° 41' 32", to find the 
angle A. 

Making b the middle part, the others will be the adjacent parts, and, 
therefore, by the rule rad. X sia. b = tan. a X tan. comp. A, 

1. • J • I, A A T^ad- sin- b , . , . 

that IS, rad. sm. b = tan. a cot. A .'. cot. A = ; and as sm. b is 

tan. a 

positive, and tan. a negative, cot. A will be negative, and, therefore, A 

will be obtuse, or the supplement of the angle given by the tables, 
rad. . . . 100000000 

tan. a 116° 30' 43" . 103020371 

sin.* 29 41 32 . . 96949041 



cot. A 103 52 48 . 9-3928670. 

4. In a spherical triangle, right-angled at C, are given b = 29° 12' SC", 
and B = 37° 26' 21", to find the side a. 



QUADRANTAL TRIANGLES. 55 

Taking a for the middle part, the other two will be adjacent parts; 

hence, by the rule, rad. X sin. a = tan. h X tan. comp. B 

n • , -r. tan. h cot. B 

that IS, rad. sin. a = tan. b cot.B .-. sm. a =^ -, 

' rad. 

In this case there are two solutions, viz. a and the supplement of a, 
both of which have the same sine. As sin. a is necessarily positive, b 
and B must necessarily be always of the same species, that is, either 
both acute or both obtuse, so that, as observed at p. 53, the sides in- 
cluding the right-angle are always of the same species as the opp. angles, 
a circumstance which must be attended to in framing examples, 
rad. . . . 100000000 

tan. h 29° 12' 50" . 9' 7475666 

cot. B 37 26 21 . 10' 1159745 




sin. a 46° 55' 2' or 133° 4' 58" . 9-8635411. 

It appears, therefore, that there exists two right-angled 3' 
triangles, having an oblique angle, and the opposite side 
in one equal to an oblique angle and the opposite side iUi 
the other, but the remaining oblique angle in the one the ' 
supplement of the remaining oblique angle in the other. 
These triangles are situated, with respect to each other, 
on the sphere, as the triangles ABC, AB'C, in the annexed c3 ^B 
diagram, in which, with the exception of the common side, AC, and the 
equal angles B, B', the parts of the one triangle are supplements of the 
corresponding parts of the other. 

5. Given the angle A = 23° 28', the side b = 49° 17', to find the hypo- 
tenuse c. c = 51° 42' 37". 

0. Given the hypotenuse c — ^^"^ 32', the side a — 37° 48', to find the 
angle B. B = 70° 19' 18". 

7. Given the perpendicular sides a = 59° 38' 27", b = 48° 24' 16". to 
find all the other parts, c = 70° 23' 42", A = 66° 20' 40", B = 52° 32' 55". 

8. Given b = 121° 26' 25", and the opposite angle B = 111° 14' 37", to 
find all the other parts. 

Solution of Qiiadrantal Triangles. 
(54.) The rules for right-angled triangles will serve also for the solu- 
tion of quadrantal triangles^ or those in which one side is a quadrant. 
For by changing such a triangle for its supplemental triangle, we shall 
then have to consider a right-angled triangle, of which the hypotenuse 
will be the supplement of the angle opposite the quadrantal side, the two 
perpendicular sides supplements of the other two angles of the proposed 
triangle, and the two oblique angles of the new triangle supplements of 
the oblique sides of the primitive triangle. That is, the sides of the pri- 
mitive or quadrantal triangle being «, b, and c = 90° and its angles A, B, 
C, the sides of the supplemental triangle will be 180° — A, 180° — B, and 
180° — C, this latter being the hypotenuse ; and the opposite angles will 
be 180° — a, 180° — b, and 90°. But the parts of a quadrantal triangle 
may be determined without the aid of the supplemental triangle. Thus 
let AD be the quadrantal side in the triangle ABD. Produce d 
DB, if necessary, till DC becomes a quadrant, and draw the 
arc AC, which will, obviously, measure the angle D, since D 
will be the pole of the arc AC, and C will be a right angle : also 
the angle CAB will be the complement of the angle BAD in 
the proposed triangle, and the angle ABC will either be iden- 
tical with ABD in the proposed, or supplemental to it, accord- 
ingly as DC exceeds, or falls short of, a quadrant; hence all. < 
the parts of the proposed triangle are easily determined from 
those of the right-angled triangle ABC. 




56 



SPHERICAL TRIGONOMETRY. 



If the angle DAB is less than 90°, or than the angle DAC, the side DB 
must, obviously, be acute ; but if DAB is greater than 90=, DB will be 
obtuse, and conversely. Hence the angles adjacent to the quadrantal 
side are of the same species as the sides opposite to them. The same 
may be inferred from the polar triangle. 

It must be remarked that the solution "will be ambiguous whenever the 
determination of the right-angled triangle becomes ambiguous, whether 
we employ the polar triangle or the triangle ABC in the above diagram. 
This ambiguity occurs only when the given parts in the right-angled tri- 
angle are one of the perpeiidicular sides and the angle opposite to it. (See 
solution, p. 54.) 

EXAMPLES. 



In the triangle DAB, DA = 90°, A = 54° 43', 
and D = 42° 12', required the other parts. 

As the angle DAB is less than 90°, that is, less 
than the angle DAC, DB is less than a quadrant, 
and, therefore, the right-angled triangle ABC is 
situated as in the figure, BC being the prolongation 
of DB. Of the parts of this right-angled triangle 
we have given A = 90° — 54° 43' = 35° 17', and b — 
42° 12', to find the other parts. 

Let A be the middle part, then i and c will be 
adjacent parts, therefore, rad. X sin. comp. A = 
tan. b X tan. comp. c, 




tliat is, rad. cos. A = tan. b cot. 



cot. c 



rad. COS. A 
tan. b 



rad. 


- 


cot. A 


35° 17' 


tan. b 


42 12 



48° C ^' 



10-0000000 
9-9118528 
9-9574850 

9-9543678. 



rad. 


_ 


COS. b 


42° 12' 


sin. A 


35 17 



Let B be the middle part, then A, b, -will be opposite parts, and. con- 
sequently, rad. X sin. comp. B == cos. b X cos. comp. A ; 

T-. 7 • A -,-. cos. b sin. A 

that IS, rad. cos. B = cos. b sm. A .'. cos. B = 

rad. 

10-0000000 
9-8697037 
9-7616424 

cos. B 64° 39' 55" - - 9-6313461. 

hence the angle ABD is 115° 20' 5" 

It remains now to find a ; let, therefore, B be the middle part, then a 
emd c will be the adjacent parts; hence 

rad. X sin. comp. B = tan. a X tan. comp. c ; 
, . J _ rad. cos. B 

that IS, rad. cos. B = tan. a cot. c .'. tan. a =- 



cot. c 



rad. 
cos. B 
cot. c 



10-0000000 
9-6313461 
9-9543678 



tan. a 25° 25' 20" - 9-676978o ; 

therefore, the side DB, which is the complement of this, is 64° 34' 40-". 

2. In the triangle DAB, DA = 90°, A == 112° 2' 9", and AB = 67= 
3' 14", to find the other parts. 

Since in this example A is obtl^se, DB is obtuse. 



OBLIQUE-ANGLED TRIANGLES. 57 

In the right-angled triangle ABC we have A = 22° 2^9'' and AB 
= 67° 3' \M' ; let A be the middle part, then AB, AC, will be adjacent 
parts, and we shall have 
rad, X sin. comp. A = tan. l X tan. comp. c ; 
that is, rad. cos. A = tan. h cot. c 
rad. COS. A 

.•. tan. = 

cot. c 
rad, . - - - - 100000000 

COS. A 22° 2' 9'' - 9-9670560 

cot. c 67 314 - - 9-6267152 




tan. b 65 27 9 - 10-3403408 ; -^ 

therefore, the angle D = 65° 27' 9'\ ^"^B 

Take now a for the middle part, then A and c will be opposite parts ; 
hence rad. X sin. a — cos. comp. A X cos. comp. c, 

, . . , . . sin. A sin. c 

that IS rad; sm. a — sm. A sm. c .-. sm. a = ; 

rad. ' 

and a will be acute, because the opposite angle is acute 

rad. - - 100000000 

sin. A 22° 2' 9'' - - 9-5742471 

sin.c 67 314 - - - 9-9641992 



sin. « 22 12 44 - - 9-53844635 

therefore BD = 110° 12' 44''. 

As we have now to find B, take a for the middle part, then h and B 

Will be adjacent parts, therefore, rad. X sin. a = tan. b tan. comp. B j 

, . , -r» -,, rad. sin. a. 
that IS, rad. sin. a = tan. b cot. B .*. cot. B = ^— 

tan. b 

rad. . . , 100000000 

sin.« . . . 9-5384463 

tan.* . . . 10-3403408 



cot. B 81°1'58" . 9-1981055. 

3. Given the quadrantal side and the other two sides equal to 22° 53' 
30", and 51° 4' 35", to find the angle opposite to the quadrantal side. 

B = 70° 3' 44". 

4. In the quadrantal triangle ADB are given D = 69° 13' 46", and 
A = 72° 12' 4", to determine the other parts. 

AB = 70° 8' 39", BD = 73° 17' 29", B = 96° 13' 23". 

These examples will suffice for the present, to show the application 
of Napier's rules to the solution of right-angled and quadrantal triangles. 
We shall, therefore, now give examples of the solution of the various 
cases of oblique-angled triangles in general. 

Solution of Oblique Angled Spherical Triangles. 

(55.) The fundamental equations (A) show that in order to deter- 
mine the several parts of a spherical triangle, three of those parts must 
be previously given. Now, three parts out of the six can be combined 
only in these different ways, viz. 

1. The three sides. 

2. The three angles. 

3. Two sides and the included angle, 

4. Two angles and the interjacent side, 

5. Two sides and an opposite angle. 

6. Two angles and an opposite side. 

H 



<^S SPHERICAL TRIGONOMETRY. 

So that the complete solution of an oblique-angled spherical triangle 
presents six cases. These we shall solve in the order in which they are 
here enumerated. 

Case i. (56.) Given the three sides to find the angles. 
For the determination of any angle A we have by (47) the three fol- 
lowing different expressions, viz. 

sin. I A 
cos. J A 

tan. i A 

We may apply to these formulas the remarks made at (21) in the 
Plane Trigonometry. It will be sufficient to observe here that the 
first formula is generally the most suitable, because the angle A is 
rarely so large as to be very near 180°. 

EXAMPLES. 

1. In an oblique spherical triangle the three sides are a = 68° 46' 2" 
b = 430 37' 38", c = 37° 10' ; required the angle A. 
a 68° 46' 2" 
sin. b 43 37 38 arith. comp. 01611739 
sin. c 37 10 arith. comp. 0-2188656 





sin 


(*s- 


- b) sin 


as- 


-0 


^ 




sin. b sin 


c 






sin 


iSsin 


(^s- 


a) 




-J 




sin. 6 


sin.c 






= ^ 


sin 


(iS- 


- b)sm. 


as- 


c) 




3in.- i S 


sin. a 


S — a) 





2)149 33 40 
74 46 50 



sin. (iS — *) 31 9 12 - - 97137678 



5in. a 

sm. (i 



sin. (iS — c) 37 36 50 - - 97855698 



2)198793771 



sin. I A 60 29 53 - - 99396885 
.-. A =. 120° 59' 46". 
3. Given a = 108°, * = 52° 12', and c = 74° 30', to find A. 
a 108° 0' 



sin. b 37 48 


arith. 


comp. 0-2126054 


sin. c 74 30 


arith. 


comp. 0-0160895 


2)220 18 




110 9 






sin. as — b) 72 21 
sin. as — c) 35 39 


_ 


9-9790594 




9-7655436 




2)19-9732979 


sin, 1 A 75 51 56 


. 


9-9866489 


.-. sin. A - 


= 1.51° 


43' 52". 


3. Given a = 70° 4' 18", b = 63° 


2T27 


", and c = 59° 16' 23", to find 


the angles A and B. 


A --. 


= 81° 38' 20", B = 70° 9' 38". 


4. Given a = 67° 25' 2", b=^80 


° 2' 25 


", c=23° 27' 46", to find the 


angle A. 




A = 54° 55' 19". 


5. Given a = 61'^ 32' 12", b = 83^ 


^19' 43 


", c = 23° 27' 46", to find A. 
A = 20° 39' 48" 



OBLIQUE-ANGLED TRIANGLES. 59 

Case n. (57.) Given the three angles to find the sides. 

By (49) we have the following formulas for any side a in tenns of the 

^, 1 • • , J — cos. \ S cos. {\ S— A) 

three angles, viz. sm. \ a=^ A : — - — r-~ - 

* ' 5 M gjn. B sm. C 

|cos.(^S— B)cos.aS— C) 

cos. i <J= J — : ~-. ^j ^ 

^ ^ sm. B sm. C 



tan. I a ■ 



I —cos. § S cos (i S — A) 



cos. (iS — B)cos.US — C). 
Jt may be remarked here that the first two only of the expressions in 
this and in the former case need be borne in the memory, as the third is 
an immediate consequence of them. If the expressions in the former 
case be recollected, these can scarcely fail to be recalled at the same 
time, as they differ from them only in this, viz. that the sides are replaced 
by their opposite angles, and, except in the denominators, cosines are 
written for sines, and sines for cosines. 

EXAMPLES. 

1. The three angles of a spherical triangle are, A = 130° 3' IT', 
B = 31° 34' 26", C = 30° 28' 12", required the side a. 
A 130° 3'11'' 
sin. B> 31 34 26 arith. comp. 0-2810023 
sin. C' 30 28 12 arith. comp. 0-2949174 



2)192 

COS. i S 96 

cos.(iS — A) 34 


5 49 
2 54i 

I6i 

1 U 

, a = 70° 


2' 3'r 


. 90227162 
9-9185570 


sin. ^ a 35 


2)19-5171929 

9-7585964 



2. The three angles of a spherical triangle are, A = 103° 59' 57'', 
B = 46° 18' 7", C = 36° T 52''; required the side a. 

a =42° 8' 48". 

3. The three angles of a spherical triangle are 120° 43' 37", 109° 55' 
42", and 116° 38' 33' ; required the three sides. 

115° 13' 26", 98° 21' 40", and 109° 50' 22." 

Case m. (58.) Given two sides a, b, and the included angle C, to 
find the other parts. By Napier's analogies, 

cos. hi^+b)- COS. i(a^b) :; cot. ^ C : tan. | (A -f B) 
sin. i{a-\-b): sin. h {a -^b) :: cot. | C : tan. i {A ~ B). 
These serve to determine the angles A, B, opposite to the given 
sides; after which the third .side c may be determined by either of the 
remaining two analogies of Napier, viz. 

COS. I (A + B) : cos. HA -^B) :: tan. * c : tan. i {a-\-b) 
sin. J (A -f- B) : sin. | (A ^B) :: tan. ^ c : tan, i {a — b). 

EXAMPLES. 

1. In a spherical triangle are given a = 38° 30', b — 70°, and C =! 
31° 34' 26", to find the other parts. 



60 SPHERICAL TRIGONOMETRY. 

I To find A and B. 

COS. i{a^b) 540 15'ar. comp.0-2334015 ar. comp. sin. 0-0906719 

COS. i (a -V J) 15 45 . 9-9833805 sin. 9-4336746 

cot. I C 15 47 13 10-5486352 105486352 



tan. J (A+B)80 15 41 10-7654172, tan. ^(a-B) 49° 47 30'' 10-0729817 
HA -j- B) must be acute, because H« + .^) is acute. 
By taking the sum and difference of these results we have, B = 130° 
3' \V\ and A = 30° 28' 11". 

n. To find c. 

COS. i (A -».B) 49° 47' 30" arith. comp. 01900575 
COS. ^ (A + B) 80 15 41 . . . 9-228-2812 
tan. Ha + ^>) 54 15 0* , 10-14-27296 

tan.lc 20 95610683 

.-. c = 40o0'0". 
When in the case we are considering, the only part required happens 
to be the side opposite the given angle, the finding of the other two angles 
then becomes merely a subsidiary operation, and the determination of 
the required side, by Napier's analogies, seems somewhat lengthy. 
But a shorter method of solution is deducible from the fundamental 
formula, cos. c = cos. a cos. b -|- sin. a sin. b cos. C . . (1). 

For substituting cos. a tan. a for its equal sin. a it becomes 
COS. c = cos. a (cos. b -\- tan. «sin. b cos. C). 

A ^ cos. 0) 

Assume lan. a cos. C = cot. w = — : 

sm. u 
^, sin. w cos. ^ + sin. & cos. u cos. a sin. (u + i") 

^nen cos. c = cos. a 7^ — '■ == ^^ — ' — ^ • 

sin. CO sin. oj 

Hence, to find the side c, we must first determine a subsidiary angle 
<o from the equation cot. w = tan. a cos C . (2) ; after which c is found 
, , ^. cos. a sin. (oj + b) 

by the equation cos. c =-■ ■. — ^^ — ! ^ - - - (3). 

sm. cd ' 

2. The same parts being given as in the last example, to determine c 
by these formulas, 

tan. a 38° 30' 0" . 99006052, cos. a . . 98935444 
cos. C 31 34 26 . 99304221, sin. co, ar. comp. 00820652 

cot. 55 52 301 . 9-8310273, sin. (o) 4- &) 99086437 

COS. c 40° . 9-8842533. 
Other formulas for the determination of c might be easily deduced 
from the same equation (1), but this is as short and as convenient as 
any. We might also introduce here a distinct formula for the deter- 
inination of one of the angles A, by help of a subsidiary arc w ; but as 
little or nothing would be gained, in point of brevity, over the process 
by Napier's analogies, we shall not stop to investigate it. 

* There will be no necessity to refer to the tables for the tangent of 
this arc, we shall obtain it by subtracting the right-hand arithmetical 
complement in the preceding logarithmic process'from that on the left, 
addmg 10 to the index. For calling the right-hand complement p, and 
the left q, and recollecting that log. tan. == 10 -j- log. sin. — log. cos. = 
10 -f (10 —p) — (10 — q), we have losr. tan. = lO-hq—p. 



OBLIQUE ANGLED TRIANGLES. 61 

3. In a spherical triangle are given a = 43° 37' 38'', h — 37° 10', 
C = 120° 53' 46", to find the side c. c = 68° 46' 2". 

4. In a spherical triangle are given the two sides, equal to 37° 10' and 
68° 46' 2", and the included angle equal to 39° 23' ; required the other 
two angles. 33° 45' 3" and 120° 59' 49". 

5. Given the two sides equal to 44° 13' 45" and 84° 14' 29", and the 
included angle equal to 36° 45' 28" ; to find the other parts. 

The angles are 32° 26' 6", and 130° 5' 22", and the side 51° 6' 12", 

Case iv. (59.) Given two angles A, B, and the interjacent side c, to 

find the other parts. 
The solution of this case as well as the former, is comprehended in 

Napier's analogies ; the one pair, viz. 

cos. I (A + B) : cos. KA -^ B) : : tan. I c : tan. | («-f b) 
sin. I (A + B) : sin. I (A -w B) : : tan. J c : tan. h{a — b)\ 

determining the unknown sides a, b, and either of the otlxer pair, viz. 
cos. i{a-\-b): cos. ^ (a-^b) :: cot. h C : tan. J ( A -f- B' 
sin. i (« -f- Z>) : sin. h {a-^b) ■ : cot. J C : tan. ^ A — B] 

enabling us to find the unknown angle C. 

EXAMPLES. 

1. In a spherical triangle are given two angles equal to 39° 23' and 
33° 45' 3", and the interjacent side equal to 68° 46' 2" ; to find the re- 
maining parts. 

I. To find the Sides. 
cos. J (A-f-B) 36° 34' H"ar. comp.00951980 ar. comp. sin. 0-2249260 

cos-KA— B) 2 48 58J - - 9 9994752 sin. 8-6913737 

tan.ic 34 23 1 - - - 98352429 98352429 



tan. J (tt + i) 40 23 49 - - - 9-9299161,tani(a-6)3°13'48"8-7515426 
.-. a = 43° 37' 37", b = 37° 10' 1". 

n. To find the Angle. 
sin. i{a — b) 3° 13' 48" arith. comp. 1-2491502 
sin.k{a-{-b) 40 23 49 9-8116281 

tan. HA — B) 2 48 58^* 8-6918985 

cot.lC 60 29 53 9-7526768 

.-. C =- 120° 59' 46". 
If the angle opposite to the given side be the only part required, a 
more compendious method of solution may be obtained by introducing 
a subsidiary arc, as in last case. Thus the formula (B) art. (50) be- 
comes when cos. A tan. A is substituted for sin. A, 

cos, C = cos. A (tan. A sin. B cos. c — cos. B) ; 

. cos. w 

or assummg tan. A cos. c = cot. m 



cos. C = CDS. A 



sm. w 
sin. B cos. w — sin. w cos. B cos. A sin, (B — w) 



Hence, having found a subsidiary angle w, by the equation 

cot. o) = tan. A cos. c (1) ; the sought angle is determined by 

, . ^ COS. A sin. (B — w) 

the equation cos. C = : -. 

sm. w 

' The log. tangent of this arc will be equal to log. sin. — log. cos., before given, in- 
creased by 10. 

6 



02 SPHERICAL TRIGONOMETKT. 

2. The given quantities being the same as in last example, to deter- 
laine the angle C. 
tan. A 393 23' 0" 99143020 cos. A 9-8881335 

COS. c 68 46 2 9o5889T9 sin. w, ar. comp. 0-0183921 

cot. w 73 26 33i 9-4731999 sin. (B — a>) 39^ 41' 30^' 9-8052682 



COS. 59=3 0'13" 9-7117938. 
As (B — w) is negative, cos. C must be negative ; hence C is the sup- 
plement of this, viz. 1-20=' 59' 47". 

3. Given A = 30= 28' 11", B = 130= 3' 11", and c = 40= : to determine 
the other parts. a = 38= 30', h = 70=, and C = 31= 34' 26". 

4. Given A = 31° 34' 26", B = 30= 28' 12", and C = 70= 2' 3" ; to find 
the angle C. C = 130= 3' 11". 

5. Given A = 34= 15' 3", B = 42= 15' 13", and C = 76= 35' 36" ; to find 
a and b. a = 40= 0' 10", 6 = 50= 10' 30". 

6. Given A = 51= 30', B = 131= SC, and c = 80= 19' 12" ; to find C. 

C = 59= 15' 59". 

Case v. (60.) Given two sides a, b, and the angle A opposite to a; to 
find the other parts B, C, c. 

1. To find the angle B vre have, by (46) the proportion, 

sin. a : sin. b :: sin. A : sin. B . . . . (1). 

2. To find C and c, vre have, by Kapier's analogies, 

cos. i (a-^^b) : cos. i (a-^b) : : tan. J (A 4- B) : cot. | C ) /n\ 
cos. i(A^B): cos. KA+B) :: tan. i {a-\-b) : tan. h c S " ^^' 
Or after either C or c is formd by one of these analogies, the other 
part may be found by the proportion sin. A : sin. C :; sin a : sin. c (3); 
although vre shall prefer Napier's analog}^ to this in order that all ambi- 
guity may be avoided. 

If only one of the parts C, c, he required, then it -vrill be best to find 
first the" angle B, by the proponion (1), ^rhich operation must be re- 
garded entirely as subsidiary to the determination of the required part, 
by one of the analogies (2). " The part determined by the proportion (1) 
admits of a double value, since two arc? answer to the same sine ; it be- 
comes necessary, therefore, for us to inquire under what circumstances 
both these values are admissible, and how we may know which to choose 
when but one solution exists. Referring to the" fundamental formula 

,.. - „ COS. i — COS. a COS. c .... 

(A), we have cos. B = -. -. : m which expression we 

sm. a sm. c 

may remark that if cos. b is numerically greater than either cos. a or 
cos. c, the second member must take the sign of cos. b, consequently, B 
and b must be of the same species if sin. b <^ sin. a, or sin. b < sin. c, 
that is, an angle must be of the same species as its opposite side, if the sine 
of this side is less than the'sine of either of the other sides. But if cos. b is 
numerically less than cos. a. then whether the right hand member be 
-}- or — will depend upon the magnitude of cos. c, or cos. c will have two 
values corresponding to -{- cos. B. and — cos. B ; hence an angle has two 
values, v:hen the sine of its opposite side is greater than the sin£ of the 
other given side. 

EXAMPLES. 

1. Given the side a = 63= 50', the side b = 80= 19', and the angle A = 
51= 30' ; to determine the other parts. 



OBLiaUE ANGLED TRIANGLES. 

I. To find the Angle B. 

sin. a 63° 50' arith. comp. 0-0469582 
sin. J 80 19 - - 9-9937679 
sin. A 51 30 - - 98935444 



: sin. B 59° 15' 47" - 99342705 
The angle B admits of two values, because sin. b > sin. a, so that 
there exist two triangles, having the data proposed. 
We shall, however, take the acute value of B. 



E 


[. To find the 


Angle C. 




COS. ^{a-^ b) 
: COS. ^ (« + *) 
::tan. KA+B 


80 14' 30" 
72 4 30 
) 55 22 53 

65 44 53 
.-. C = 131^ 


arith. comp 
29'"46"." 


. 00045086 

9-4882288 
101609412 


: cot. k C 


9-6536786 




m. To find the Side c. 




COS. i (A -w B) 
: COS. i (A + B) 
:: tan. J {a-\-b) 


30 52' 53" 
55 22 53 
72 4 30 

60 24 
2 


arith. comp 
25", and A = 


. 0-0009973 

9-7544333 

10-4901618 


: tan. \ c 


10-2455924 


.-. C - 
2. Given a = 40^ 36' : 
ine C. 


= 120 48 0. 
57", 6 = 91^3' 


= 350 57' 15" 


I. To find the siobsidiary Angle 
sin. a 40° 36' 37" arith. comp. 
; sin. * 91 3 25 
:: sin. A 35 57 15 


B. 

0-1864788 
9-9999261 
9-7687401 



to deier- 



: sin. B 64 24 19 - - 9-9551450, 

or 115 35 41 

The angle B admits of two values, because sin. b > sin. a. We shall 
suppose the particular triangle under consideration to have B obtuse 

11. To find C. 
cos. iia'^b) 25° 13' 24" arith, comp, 0-0435177 
:cos.H«+&) 65 50. 1 - - 9-0121350 

::tan. J(A4-B) 75 46 28 - - 10-5959988 



eot. iC 29 15 28 - - 102516515 



.-. C =3 58 30 56. 

3. Given a = 40° 18' 29", b = 67° 14' 28", and A = 34° 22' 17"; to 
determine the other parts when B is acute, 

B = 53° 35' 15", C = 119° 13' 31", c = 89° 47' 6." 

4. Given a = 84° 14' 29", ^> = 44° 13' 45", and A = 130° 5' 22"; 
to determine the other parts, 

B = 32° 26' 61", C = 36° 45' 28", c = 51° 6' 12". 

5. Given a = 97° 18' 39", b = 86° 53' 46". and A = 97° 21' 26"; 
to determine c. c = 89° 21' 37". 



64 SPHERICAL TRIGONOMETRY. 

CASE VI. (61.) Given two angles A, B, and the side a opposite to one 
of them, to find the other parts. 

1. To find h we have sin. A : sin. B :: sin. a : sin. h. 

2. And to find C and c we may employ Napier's analogies, which 
need not be here repeated. 

The nature of the arc h may be discussed, as in the preceding case. 

mi T /. 1 /T^s /../^v • 7 cos. B + cos. A cos. C 

Thus the formula (B), art. (50), gives cos. h = — ^ — r-~ — : — ^ j 

sin. A sin. O 
from which it follows, as in the foregoing case, that if cos. B is nume;- 
rically greater than cos. A, B and b, will be of the same species. If 
COS. B is numerically less than cos. A, then both the values of b, given 
by the above proportion, will be admissible, for C may be determined so 
as to render cos< b positive or negative. Hence a7iy side will be of the 
same species as its opposite angle, if the sine of this angle be less than the 
sine of either of the other angles ; and the species of the sid.e b will be in- 
deferinined if the sine of its opposite angle B be greater than the sine of the 
other given angle A. There cannot, therefore, be two solutions unless 
a and A are of the same species. 



1. In an oblique-angled spherical triangle ABC are given, A = 32^ 
26' 6f ", B = 130° 5' 23", and the side a = U° 13' 42" ; to determine the 

other parts, 

I. To find the Side b. 

As sin. A 320 26' 6-|" arith. comp. 9-8836842 
: sin. B 130 5 22 - - 02705556 

: : sin. « 44 13 45 - 9' 8435629 



: sin. i 84 14 29 - - 9-9978027 ; 

b has two values, because the sine of B is greater than that of A. We 
shall take the acute value. 

II. To find the Side c. 

As COS. K A ^ B) 48° 49' 37|" arith. comp. 0-1815543 
: cos. HA + B) 81 15 441 - - 9-1815890 
:: tan. i(a + i) 64 14 7 - 103163591 

: tan. |c 25 33 6 - - 96795024 



.•.c = 51 6 12., 

III. To find the Angle C. 

As cos. i(a -^b) 20° 0' 22" arith. comp. 0-0270310 
: COS. J (a + J) 64 14 7 - - 9-6381663 
: : tan. i (A + B) 81 15 44| - 108133435 

: cot.iC 18 22 44 . . 10-4785408 

2 



.-. Cr::.36 45 28. 

2. Given A = 103° 59' 57i'^ B = 46° 18' 7i", and a = 42° 8' 48"- to find 
the angle C. e = 36or52r. 

3. Given A = 17° 46' 16^", B == 151° 43' 52", and a = 37° 48'; to find 
the remaining sides, b being obtuse. b = 108°, c — 74° 30'. 



OBLIQUE ANGLED TRIANGLES. 65 



Pre\'iously to closing this second part it may be worth while to re- 
mark, that if, in the foregoing in\restigations, we consider the radius of 
the sphere, upon which the triangles concerned are described, to be in- 
finite, then, as any finite portion of the spheric surface may be considered 
as a plane, the spherical triangles will become plane triangles, and the 
sines and tangents of their sides will become identical with the sides 
themselves ; so that all the foregoing rules and formulas, into which 
cosines, cotangents, secants, or cosecants, of the sides do not enter, are 
applicable as well to plane as to spherical triangles. 

Professor Vince, at page 43 of his Trigonometry, has the following 
note. 

" Diificulties have frequently arisen in consequence of its being sup- 
posed that an arc of 90° has a tangent and secant, each infinite. For in- 
stance, in a right-angled spherical triangle, radius : cosine of the angle 
at the base :: tangent of the hypotenuse : tangent of the base ; now when 
the base = 90°, the hypotenuse = 90° ; and, therefore, these arcs being 
equal, if they have any tangents, of whatever value they may be, they 
must be equal ; and, therefore, radius = cosine of the angle at the base, 
whatever that angle may be. This false conclusion arises from the 
supposition that an arc increases till it becomes 90° ; the tangent and se- 
cant increase without limit ; and at 90° the arc ceases to have either a 
tangent or secant, by their definition. As the arc, by increasing, passes 
through 90°, the tangent and secant increase without limit, cease to exist 
at 90°, and then begin again at a quantity indefinitely great. And thus 
in other cases where the tangent or secant of an arc enter into the com- 
putation, when the arc becomes 90°, we can draw no conclusion on 
which we can depend." 

The foregoing reasoning is very much calculated to mislead the young 
student, although it does in reality tend to overturn the author's own 
hypothesis, and to show that the tangent of 90° must necessarily be in- 
finite. 

Taking the example chosen above, by Mr. Vince, we have for the 
^ ^ J tan. 90° , . , ^ ., . 

true solution cos. < at base = rad. — — -: which must necessarily m-* 

^ tan. 90° 

volve the absurdity noticed above, except tan. 90° be either or oo ; but 
when the proper value oo is put for tan. 90°, then we have cos. < at base 

rad. ^ rad, — ; and as — admits not only of the particular value 1 

fixed upon by Mr. Vince, but of an indefinite number of values, so does 
cos. < at base. 

Upon the same grounds that Mr. Vince has rejected the tangent of 
90°, he should have rejected the cosine of 90°, which, however, he ad- 
mits to be 0. 

cos <f at vertex 

For sin. < at base = rad. — , ; but, when both base and 

cos. base 
hypotenuse are 90°, the angle at the vertex is 90°, and we ought, there- 
fore, to have, according to Mr. Vines, sin. at base = rad, which is, in- 
deed, one solution, but by no means the only one, because the values of 

-^ are innumerable. 

" I 



PART III. 



APPIilCATION OF PLANE AND SPHERICAL TRIGONOMETRY TO THE PRINCIPLES 
OF NAVIGATION AND NAUTICAL ASTRONOMY. 

(62.) Having in the two preceding parts of the present treatise pretty 
fully explained and illustrated the principles of plane and spherical 
trigonometry, we shall now, for the purpose of showing the practical 
utility of these principles, apply them to the solution of one of the most 
important mathematical problems that has ever engaged the attention 
of man, viz. to determine the place of a ship at sea. 

When a ship sails from any known place, and a correct account is 
kept of her various directions, and rates of sailing, her situation at any- 
time may he readily ascertained by the rules of plane trigonometry, and 
the solution of the problem from these data belongs to Navigation. 

But it is impossible to measure a ship's course and the distance 
sailed exactly ; so that after a long passage it would be unsafe to com- 
pute the place of the ship from the ship's reckoning. In such cases, 
therefore, the solution must be effected from other data, independent of 
the ship's account ; these are furnished by astronomical observation, and 
the computation is performed by the rules of spherical trigonometry j 
the problem then becomes one of Nautical Astronomy. We shall de- 
vote a distinct chapter to each of these important branches. 



CHAPTER I. 

THE PRINCIPLES OF NAVIGATION. 

Definitions. 
(63.) 1. The earth is very nearly spherical. For the purposes of 
Navigation it may be considered as perfectly so. It revolves round one 
of its diameters, called its axis, in about twenty-four hours. This ro- 
tation is from the west towards the east, causing the heavenly bodies 
to have an apparent motion from the east towards the west. 

2. The great circle, whose poles are the extremities of the axis, is 
called the equator. The poles of the equator are called also the poles 
of the earth ; the one being the north pole, and the other the south pole. 

3. Every great circle which passes through the poles, and which, 
therefore, cuts the equator at right-angles, is called a meridian circle. 
Through every place on the surface of the earth such a great circle is 
supposed to be drawn; it is the meridian of the place. It is expedient 
for the purposes of Geography and Navigation to fix upon one of these 
meridians as z, first meridian, from which the meridians of other places 
are measured. 

The English have fixed upon the meridian of Greenwich Observatory 
for the first meridian. 

4. The longitude of any place is the arc of the equator, intercepted 
between the meridian of that place and the first meridian ; the longi- 
tude, therefore, is the measure of the angle between the two meridians. 
The longitude is east or west, according as the place is situated on the 
right or on the left of the first meridian, when we look towards the 
north pole. 



PKINCIPLES OF NAVIGATION. 



67 



5. The difference of longitude between two places is the arc of the 
equator intercepted between the meridians of those places, or the mea- 
sure of the angle which they include ; hence, when the longitudes of the 
places are of the same denomination, that is, either boih east or both 
w^est, the difference is found by subtracting the one from the other; but 
when they are of contrary denominations the difference is found by 
adding the one to the other. 

6. The latitude of a place is its distance from the equator, measured 
on the meridian of the place. Latitude, therefore, is north or south, 
according to the pole towards which it is measured, and cannot exceed 

7. The small circles drawn parallel to the equator are called parallels 
of latitude. The arc of a meridian, intercepted between two such pa- 
rallels, drawn through any two places, measures the differences of 
latitude of those places : when the latitudes are of the same denomi- 
nation the difference of latitude is found by subtraction, but when th? 
denominations are not the same the difference of latitude is found by 
addition, like difference of longitude. 

8. The horizon of any place is an imaginary plane, conceived to 
touch the surface of the earth at that place, and to be extended to the 
heavens ; such a plane is called the sensible horizon, and one parallel to 
it, but passing through the earth's centre is the rational horizon of the 
place. A line drawn across the horizon and through the place, in the 
plane of its meridian, is the meridian of the horizon, or the north and 
south line ; the horizontal line through the same point, and perpendi- 
cular to this, is the east and west line. Besides the North, South, East, 
and West, points thus marked on the boundary of the horizon, this 
boundary is conceived to be subdivided into other intermediate points, 
corresponding to the divisions in the circle below. 

9. The course of a ship is the angle which her track makes v/ith the 
meridians ; so long as this angle remains the same, the ship is said to 
sail on the same rhumb line or loxodromic curve; The magnitude of 
the angle or the course is indicated by the mariner^ s compo.ss. 

10. The Mariner's compass consists of a circular card whose circum- 
ference is divided into thirty-two equal parts, called points, and each 
of these are subdivided into four equal parts, called quarter points ; 
across this card is fixed a slender bar of magnetized steel, called the 
needle ; the tapering extremities of which point to two diametrically 
opposite divisions of the card. These opposite divisions are marked 
N. and S., corresponding to the north and south "poles, or ends, of the 
magnetized bar. The diameter W. E., at right angles to the diam,eter 
N. S., point out the west and east points ; 
these four are called the cardinal points, 
and the others are marked as in the sub- 
joined diagram. 

Thus one point from the north towards 
the east is north by east ; two points, north, 
north east ; three points, north-east by north ; j 
and so on. (See the table of Rhumbs at ' 
the end, p. 300.) 

The card thus furnished being now s ap- 
pended horizontally, and so as to allow the 
needle to settle itself freely, will point out 
the four cardinal points of the horizon, as 
also the several intermediate points, provided only that it is the property 
of the magnetic needle to point due north and south. Such, however, 
is not strictly the case, as the needle is found from accurate observa- 
tions, to deviate from this position, and at some places very consider- 
ably, and this deviation is itself subject to variation. But the true 




OS PLANE AND SPHERICAL TRIGONOMETRY. 

direction of the compass, or the angle it makes at any place with a line 
pointing duly north and south, maybe ascertained at any time by astro- 
nomical observations, and thus the deviation of the compass-points, 
from the corresponding points of the horizon, may always be found and 
allowed for. 

The compass is so placed on ship-board that the vertical p)lane, 
cutting the ship from stem to stern, may pass through the centre of the 
card, so that that point of the compass which is directed to the ship's 
head shows the coynpass-course, and the proper correction for variation 
being applied the true course will be obtained. 

11. A ship's rate of sailing is determined by means of an instrument, 
called the Log, and an attached line called the log-line. The log is a 
piece of wood forming the sector of a circle, audits rim is so loaded 
with lead that when heaved into the sea it assumes a vertical position, 
with its centre barely above the water. The log line is so attached as 
to keep the face of the log towards the ship, that it may offer the greater 
resistance to be dragged after the ship hy the log-line, as it unwinds 
from a reel on board, by the advancing motion of the ship. The 
length of line thus unwound in half a minute, gives the rate of sailing.- 
For convenience the log-line is divided into equal parts, called knots, of 
,which each measures the 120th of a nautical or geographical mile,* and 
as half a minute is the 120th of an hour, it follows that the number of 
knots, and parts of a knot, run in half a minute expresses the number 
of miles and parts of a mile, run in an hour, at the same rate of 
sailing. 

On Plane SaAling. 

(64.) Let the annexed diagram represent a portion of the earth's surr 
face, P being the pole, and EQ, the equator. 
Let AB be any rhumb line, or track described ^ — "^^^^ 

by a ship in sailing on a single course from A 
to B. Conceive the path of the ship to be di- 
vided into portions Ab, be, cd, &c. so small 
that each may differ insensibly from a straight 

line, and draw meridians through these seve- / fJ/lPFDJ^^ 

ral divisions, as also the parallels of latitude 3^ 
hh',cc', dd/, &c. ; we shall thus have a series of 
triangles described on the surface of the globe, but so small that each 
may be considered as a plane triangle. These triangles are all similar, 
for the angles at b', c', d/, &c. are right angles, and the ship's path cuts 
all the meridians at equal angles; hence (Geom. prop. 9, Book 6.) 

Ab : Ab' \\bc : be' :: cd • cd', &c. therefore, (Geom. prop. 5, Book5,) 
Ab ■.Ab''.\Ab-\-bc-^cd-{-&.c. : Ab' ^bc' -^cd' ^&ic. 

But Ab-\-bc-\- cd -j- &c. is the whole distance sailed, and Ab' -\- be' 
+ cd/ -f &c. — AW, is the difference of latitude between A and B ; 
consequently, if a right-angled triangle ABB', similar to the small tri- 
angle Abb', be constructed, that is, one in which the angle A is equal 
to the course, and if the lu-potenuse AB represent the dis-g jg 

tance sailed, the side AB'will represent the difference 01 "^ 
latitude. Moreover, the other side BB', or that opposite to 
the course, will represent the sum b'b -f c'c -\-d'd +&c. of all 
the minute departures which the ship makes from the succes- 
sive meridians which it crosses ; for as the triangle ABB', in 
this last diagram, is similar to the small triangle Abb', in the 
former, we have Ab : bb' : : AB : BB' . . (1); 
but in the first figure we have Ab : bb'y.bc : cc':: cd : dd'. &c. 

.'.Ab: bb' : : Ao -f be' -|- cd -j- &c. : bb' -j- cc' -f dd' -{- &.c. 

The geographical mile is one minute of the earth's circumference. Taking the 
diameter at 7916 English miles, the geographical mile will be about 6079 feet. 





PRINCIPLES OF NAVIGATION. 



consequently, since the three first terms of (1) are respectively equal to 
those of (2), the remaining terms BB\ bb' -{- cr/ -j- dd' -\- &c. m\xsi he 
equal. This last quantity is called the departure of the ship in sailing 
from A to B. It follows, therefore, that the distance sailed^ the difference 
of latitude made, and the departure, are correctly represented by the liypo- 
tenuse and sides of a right-angled plane triangle, in which the angle oppo- 
site the departure is the course, so that when any two of these four things 
are given the others may be found simply by the resolution of a right- 
angled plane triangle ; as far, therefore, as these particulars are con- 
cerned the results are the same as if the ship were sailing on a plane sur- 
face, the meridians being parallel straight lines, and the parallels of 
latitude cutting them at right-angles ; and hence that part of Navigation 
in Vv-hich only distance sailed, departure, difierence of latitude, and 
course are considered, is called Plane sailing. 

EXAMPLES. 

1. A ship from latitude 47° 30' N. has sailed S. W. by S. 98 miles, 
What latitude is she in, and what departure has she 
made'? 

Let C be the place sailed from, CB the meridian, 
the angle 0=3 points = 33^ 45' and C A = 98 miles, 
the distance sailed ; then CB will be the difierence of 
latitude, and BA the departure 



As rad, 
: Distance 98 
: : cos. course 33° 45' 



10- 
1-9912261 
9-9198464 



As rad. 
: Dist. 
: ; sin, course 




: Diff. of lat. 81-48 1-9110725 : Departure 54-45 1-7359651 

Latitude left 47° 30' N. 
Diff". of lat. = 81-48 minutes = 1° 22' S. Dep, = 54-45 miles W. 
Latitude in 46 8 N. 
2. A. ship sails for M hours on a direct course, from lat. 38° 32' N., 
till she arrives at lat. 36° 56' N. ; the course is between the S. and E., 
and the rate 5| miles an hour. Required the course, distance, and 
departure. 

Lat. left 38° 32' N. 24 X 5| = 132 miles, the distance. 

Lat. in 36 56 N. 



Difi". 
As Dist. 
: Rad. 
: : Diff: lat. 



1 36 
132 



96 



= 96 miles. 
21205739 
10- 

1-9822712 



As rad. 
: Dist. 
: : sin. course 



10- 

21205739 
9-8364771 



: cos, course 43° 20' 9-8616973 : Dep. 9058 19570510 

Hence the course is S. 43° 20' E., and the departure 90-58 miles E. 

3. A ship sails from lat 3° 52' S. to lat. 4° 30' N., the course being 
N. Vv. by W. 2 W, ; required the distance and departure. 

Distance 1065 miles. Departure 938'9 miles W. 

4. Two ports lie under the same meridian, one in latitude 52° 30' N., 
and the other in latitude 47° 10' N. A ship from the southernmost sails 
due east at the rate of 9 miles an hour, and two days after meets a 
sloop which had sailed from the northernmost port ; required the sloop's 
direct course and distance run. 

Course S. 53° 28' E., or S. E. | E; ; the distance run 537*6 miles; 

5. If a ship from lat. 48° 27' S., sail S. W. by W., 7 miles an hour, in 
what time will she arrive at the parallel of 50° S. % In 23-914 hours. 

6. If after a ship has sailed from lat. 40° 21' N. to lat. 46° 18' N., she 
be found 216 miles to the eastward of the port left ; required her course 
and distance sailed. Course N. 31° 11' E., dist. 417-3 miles. 



70 PLANE AND SPHERICAL TRIGONOMETRY. 

Traverse Sailing. 

(65.) When a ship in going from one place to another, sails on dif- 
ferent courses, it is called traverse sailing ; and the determination of 
the single course and distance from the one place to the other is called 
working or compounding the traverse. To effect this, it is obviously 
merely necessary to find the difference of latitude, and departure, due 
to each distinct course, to take the aggregate of these for the whole dif- 
ference of latitude and departure, and from these to find, as in last 
article, the single course and distance. It is usual in thus compound- 
ing courses to form a table consisting of six columns, called a tro^verse 
table, and in the first column to register the several component courses, 
and against them, in the second column, the proper distances ; the 
next two columns, marked N. and S., are to receive the several dif- 
ferences of latitude, whether N. or S., due to each course, and dis- 
tance, and the two remaining columns marked E. and W. are to re- 
ceive, in like manner, the corresponding eastings and westings, that is, 
the departures. When these several particulars are all inserted, the 
columns are added up, and the difference of the results of the N. and S. 
columns will be the required difference of latitude, and the difference 
of the results of the E. and W. columns will be the corresponding de- 
parture. 

The columns appropriated to the differences of latitude and depart- 
ures are usually filled up from a table already computed to every 
quarter point of the compass, and to all distances from one mile up to 
a hundred or 120 ; so that, by entering this table with any given course 
and dista.nce, the proper difference of latitude and departure is found 
by inspection. Most books on navigation contain also a second and 
more enlarged traverse table, being computed to every course from a 
quarter of a degree up to forty-five degrees. This latter table we have 
not thought it necessary to insert in our collection, but the former we 
have given (Table iv.), and its use is fully explained in the introduction 
prefixed. 

But there is another mode of finding the direct course and distance, 
much practised by seamen, viz. by construction. To facilitate this con- 
struction the mariner's scale is eniployed, which is a two-foot flat rule 
exhibiting several scales on each side, by help of which and a pair of 
compasses the usual problems in sailing may be all solved. One of 
these scales is a scale of chords, commonly called a scale of rhumbs, 
being confined to every quarter point of the compass ; and another is a 
more enlarged scale of chords, being to every single degree. Both these 
scales are constructed in reference to the same common radius, so that 
the chords on the scale of rhumbs belong to that circle whose radius 
equals the chord of 60° on the scale of chords ; and the method of lay- 
ing down a traverse from these scales, and one of equal parts, and of 
thence measuring the equivalent single course, and distance made good, 
will be at once understood from the following examples, 

EXAMPLES. 

1. A ship sails from a place in lat. 24° 32' N., and has run the fol- 
lowing courses and distances, viz. 

1st, S. W. by W., distance 45 miles; 2d, E. S.E., distance 50 miles; 
3d, S. W., distance 30 miles ; 4th, S. E. by E., distance 60 miles ; 5th, 
S. W. by S. i W., distance 63 miles : required her present latitude, with 
the direct course and distance from the place left to the place arrived at. 



PRINCIPLES OF NAVIGATION. 



71 





Traverse 


Table. 






Courses. 


Dist. 


Difference of Lat. 


Departure. 






N. 


S. 


E. 


W. 


S. W. byW. 


45 




250 




37-4 


E.S.E. 


50 




191 


46-2 




s. w. 


30 




21-2 




21-2 


S. E. by E. 


60 




33-3 


49-9 




S. W. by S. i W. 


63 




50-6 




37-5 








149-2 


961 


961 



It appears from the results of this table that the difference of latitude 
made by the ship daring the traverse is 149'2 S. = 2° 29' S. 
Lat. left - . - . 24° 32' N. 

Diff. lat. - . = . 2 29 S. 



Lat. in 



22 3 N. 




It appears also that the departures east are equal to the departures 
west, so that the ship has returned to the meridian she sailed from, con- 
sequently the direct course from the place left to that come to is due 
south, and the distance is equal to the difference of latitude, which is 
149-2 miles. 

The construction of this traverse is as follows. 

With the chord of 60°, taken from the line of chords 
on the mariner's scale, describe the horizon circle, and 
draw the north and south line N. S. From the line of 
rhumbs take the chords of the several courses, and as 
these are all southerly, they must be laid off from the 
south point S, those which are westerly to the left, and j\ 
those which are easterly to the right, their extremities 
being marked 1, 2, 3, &c. in the order of the courses. 
This done, lay off from any convenient scale of equal 
parts, and in the direction Al the distance AB sailed on 
the first course ; then in the direction parallel to A2, the 
distance BC sailed on the second course ; in the direc- 
tion parallel to A3, the distance CD on the third course; 
in the direction parallel to A4, the distance DE on the 
fourth course ; and, lastly, in the direction parallel to A5, the distance 
EF on the third course ; then F will represent the plane of the ship at 
the end of the traverse ; FA, being applied to the scale of equal parts, 
will show the distance made good, and the chord of the arc included 
between this distance, and the meridian, being applied to the line of 
rhumbs, will show the direct course. In the present case the intercepted 
arc will be 0, showing that F is on the meridian of A. 

2. A ship from lat. 28° 32' N., has run the following courses, viz. 1st, 
N. W. by N., 20 miles ; 2d, S. W., 40 miles ; 3d, N. E. by E-, 60 miles : 
4th, S. E., 55 miles ; 5th, W. by S., 41 miles ; 6th, E. N. E-, m miles. 
Required her present latitude, the distance made good, and the direct 
course from the place left to that come to. 

The direct course is due east, and distance 70-2 miles, the ship being in 
the same latitude at the end as at the beginning of the traverse. 

3. A ship from lat. 41° 12' N., sails S. W. by W., 21 miles ; S. "W. ^ S. 
31 miles ; W.S.W. I S:, 16 miles ; S. | E., 18 miles ; S. W. i W., 14 
miles ; and W. | N., 30 miles : required the latitude of the place 
arrived at, and the direct course and distance to it- 

Lat. 40° 5' N. : course S. 52° 49' W. ; distance 111-7 miles. 



72 



PANE AND SPHERICAL TRIGONOXETRT. 



4. A ship from Cape Clear, in lat. 51" 25' X, sails 1st, S. S. E. i E., 16 
miles ; 2d, E.S.E., 23 miles ; 3d, S.W. by W. h W., 36 miles ; 4th, W. | N., 
12 miles ; 5th, S.E. by E. i E., 41 miles ; required the distance made good, 
the direct course, and the latitude in 1 

Traverse Table. 



Courses. 


Dist. 


Difference of Lat. i 


Departure. 






N. 


S. ; 


E. W. 


S.S.E. iE. 


16 




14-5 


6-3 




E.S.E, 


23 




8-8 


21-3 




S.W.byW.iW. 


36 




170 




31-8 


W. 1 N. 


12 


1-8 






119 


S. E. by E. \ E. 


41 




2M 


35-2 








1-8 


61-4 


63-3 


43-7 








1-8 


43-7 


. 




59-6 


19-6 



Lat. left 

Diff. lat. 59-6w 

Lat. in 
As diff. lat. 59-6 
: rad. 
:: departure 19" 6 



25^ N. 
S. 



1-7752463 
10- 
1-2922561 



As sin. cour.^e 
: departure 
:: rad 



50 25 N. 

9-4946205 
. 1-2922561 
10- 



: tan. course 18° 12' 9-5170098 : distance 6274 . 1-7976356 
therefore, as the difference of latitude is south, and the departure east, 
the direct course is S. 18^ 12' E., and the distance made good 62-74 miles. 
To construct this traverse, describe, as before, the horizon circle, with 
a radius equal to the chord of 60=, and taking from the line of rhumbs the 
chord of the first course, 2i points, apply it from S. to 1, to the right of S.N., 
as this course is south-easterly: apply, in like manner, the chord of the' 
second course, 6 points from S." to 2, also to the right of the meridian line; 
apply the chord of the third course, 5| points from S. to 3, to the left of the 
meridian, the chord of the fourth course, 7i from N. to 4, to the left of 
N. S., this course being north-westerly, and, 
lastly, apply the chord of the fifth course, b\ 
points, from S. to 5, to the right of S. N. In 
the direction Al, lay off the distance AB = 16^ 
miles from a scale of equal parts: in the direc-' 
tion parallel to AGl, lay off the distance BC = , 
23 miles; in the direction parallel to A3, lay off "^^ 
CD = 36 miles ; in the direction parallel to A4, 
lay off DE = 12 miles ; and, lastly, in the direc- 
tion parallel to A5, lay off EF = 41, then F will 
be the place of the ship at the end of the tra- 
verse ; consequently, AF will be the distance 
made good, and the angle FAS the direct course ; ^ 
applying, therefore, the distance AF to the ^ 
scale of equal parts, we shall fmd it reach 
from to 62f ; and apphang the distance Sa to 
the line of chords, we shall find it reach from 
to 18°. 

5. A ship rims the foUowins: courses, viz. 

1st, S. E., 40 miles; 2d, N.E., 28 miles ; 3d, S.W. by W., 52 miles ; 




PLANE AND SPHERICAL TRIGONOMETRY. 73 

4th, N.W. by W., 30 miles ; 5ih, S.S.E., 36 miles; Gth, S.E. by E., 58 

miles : required the direct course and distance made good. 

Direct course S. 25° 59' E., or S.S.E. f E. nearly; distance 95-87 miles. 

6. A ship in latitude 37° 10' N. is bound to a port in the latitude of 
33^ 0' N. which lies 180 miles west of the meridian of the ship ; but by 
reason of contrary winds, she sails the following courses, viz. S. W. by 
W. 27 miles, W. S. W. h W. 30 miles, W. by S. 25 miles, W. by N. 18 
miles, S. S.E. 32 miles, S. S. E. f E. 27 miles, S. E. 25 miles, S. 31 miles, 
and S. S. E. 39 miles. Required the latitude the ship is in, and her depart- 
ure from the meridian, with the course and distance to her intended port "? 

The difference of latitude and departure made on each course, will 
be seen by sketching a traverse table; hence it appears that the difference 
of latitude made good is 169-4 miles, the departure 47-4 miles, and by 
plane sailing, the course S. 15° 38' W. and distance 175'9 miles ; and 
the course to the intended port S. 58° 42' W., distance 155.-2 miles ; the 
latitude being 34° 21' N. 

These examples will, perhaps, suffice to illustrate the principles of 
plane sailing, in which, course, distance, difference of latitude, and 
departure, are the only things concerned. The determination of the 
difference of longitude made on any course cannot be effected by these 
principles, for this element is not the same as if the meridians were all 
parallel to each other, as is the case with the other elements. The find- 
ing of the difference of longitude is the easiest when the ship sails due 
east or due west, that is, upon a parallel of latitude ; this is called pa- 
rallel sailing. 

Parallel Sailing. 

(66.) The theory of parallel sailing is comprehended in the following 
proposition, viz. 

The cosine of the latitude of the parallel is to the distance run as the 
radius to the difference of longitude. This may be demonstrated as 
follows. 

In the figure, at page 42, let 7Q,H represent the equator, and BDA 
any parallel of latitude ; CI will be the radius of the equator, and cB the 
radius of the p^iallel. LetBD be the distance sailed, then the difference 
of longitude will be measured by the arc IQ, of the equator, and since 
(Geom., prop. 12, Cor. 2, B. 7) similar arcs are to each other as the 
radii of the circles to which they belong, we have 

cB : CI :: dist. BD : diff. long. la. 

ButcB is the cosine of the latitude IB to the radius CI, that is, cB is 
CI times the trigonometrical cosine of the latitude ; hence the above 
proportion is CI X cos. lat. : CI :; distance : diff. long. 

.-.COS. lat. : Rad. ( = 1) :: distance : diff. long. . . , (1). 
Corollary : hence if the distance between any two meridians, measured 
in a parallel in latitude L be D, and the distance of the same meridians, 
measured on a parallel, in latitude L' be D', we shall have, (Geom., 
prop. 15, Cor. 2, Book 5,) cos. L : D :: cos. L' : D' . . 

Hence if one of the legs of a right-angled triangle represent 
the distance run on any parallel, and the adjacent acute 
angle be equal to the degrees of lat. of that parallel, then 
the hypotenuse will represent the difference of longitude, 
since this hypotenuse will be determined by the foregoing 
proportion (1). It follows, therefore, that any problem in _ 
parallel sailing may be solved by the traverse table, com- ' disi. 
puted to degrees, as a simple case of plane sailing; for by considering 
the latitude as the course, and the distance as the difference of latitude, 
the corresponding distance in the table will express the difference of 
longitude. 

7 J 




"74 PLANE AND SPHERICAL TEIGONOMETRT. 

EXAMPLES. 

1. A ship from latitude 53° 56' N., longitude 10° 18' E., nas sailed 
due west, 236 miles : required her present longitude. 
By the rule ; As cos. lat. 53° 56' ... 9-7699134 

: radius - _ _ IQ- 

:: distance 236 - - 2-3729120 



: diff. long. 408-87 - 2-6029986 

Long, left . . 10° 18' E. 

Diff. long. == -— — degrees = 6 40 W. 

bO 



Long, in . . . 3 38 E. 

2. If a ship sail E. 126 miles, from the North Cape, in lat. 71° 10'N._, 
and then due N., till she reaches lat. 73° 26' N. ; how far must she sail 
W. to reach the meridian of the North Cape 1 

Here the ship sails on two parallels of latitude, first on the parallel of 
71° 10', and llien on the parallel of 73° 26', and makes the same differ- 
ence of longitade on each parallel. Hence, by the corollary, 
As cos. lat. 71° 10' arith. comp. 0-4910444 
: distance 126 . 2-1003705 

:: cos. lat. 73 26 . . 945504^1:1 



: distance 1113 . . 20464590. 

3. A ship in latitude 32° N., sails due east; till her difference of lon- 
gitude is 384 miles ; required the distance run. 325-6 miles. 

4. If two ships in latitude 44° 30' N., distant from each other 216 
miles, should both sail directly south till their distance is 256 miles, 
what latitude would they arrive at 1 32° 17' S. 

5. Two ships in the parallel of 47° 54' N., have 9° 35' difference of 
longitude, and they both sail directly south, a distance of 836 miles; 
required their distance from each other at the parallel left, and at that 
reached. 3855 miles, and 479-9 miles. 

Middle Latitude Sailing. 

(67.) Having seen how the longitude which a ship makes when sail- 
ing on a parallel of latitude may be determined, we come now to 
examine the more general problem, viz. to find the longitude a ship 
makes when sailing upon any oblique rhumb. 

There are two methods of solving this problem, the one by what is 
called middle latitude sailing, and the other by Mercator^s sailing. The 
first of these methods is confined in its application, and is moreover 
somewhat inaccurate even where applicable; the second is perfectly 
general, and rigorously true ; but still there are cases in which it is ad- 
visable to employ the method of middle latitude sailing, in preference 
to that of Mercator's sailing ; it is, therefore, proper that middle lati- 
tude sailing should be explained, especially since, by means of a cor- 
rection to be hereafter noticed, the usual inaccuracy of this method may 
be rectified. 

Middle latitude sailing proceeds on the supposition that the departure 
or sum of all the meridional distances b'b, c'c, p ■« 

d'd, &c. from A to B, is equal to the distance ^--— ^!5=!^^-d 

M'M of the meridians of A and B, measured 
on the middle parallel of latitude between A 
and B. 

This supposition becomes very inaccurate 
when the course is small, and the distance run 
great; for it is plain that the middle latitude^ 




I 



PRINCIPLES OF NAVIGATION. 



75 



^- 



distance will receire amuch greater accession than the departure, if the 
track of B cuts the successive meridians at a very small angle. 

The principle approaches nearer to accuracy as the angle A of the 
course increases, because then as but little advance is made in latitude, 
the several component departures lie more in the immediate vicinity of 
the middle latitude parallel. But stilt, as in very high latitudes, a small 
advance in latitude makes a considerable difference in meridional dis- 
tances, this principle is not to be recommended in such latitudes if much 
accuracy is required. 

By means, however, of a small table of corrections, recently con- 
structed by Ml-. Workman, and judiciously. introduced by Mr. Riddle in 
the second edition of his valuable Treatise on Navigation, the imper- 
fections of the middle latitude method maybe removed, and the results 
of it rendered in all cases accurate. This table we have given at the 
end of the present volume, and have explained its construction in the 
introductory explanation to ttie Tables. 

The rules for middle latitude sailing may be thus deduced. 

It has been seen at (64) that the difference of latitude, departure, and 
distance, sailed on any oblique rhumb, will be all accu- " 
rately represented by the sides AB', B'B, AB, of a plane 
triangle. Now, by the present hypothesis, tie departure 
B'B is equal to the middle latitude distance between the . 
meridians of the places sailed from, and arrived at, so that B 
the difference of longitude of the two places of the ship is 
the same as if it had sailed the distance B'B, on the middle 
latitude parallel; the determination of this difference of 
longitude is, therefore, reduced to a case of parallel sailing, for BB', 
now representing the distance on the parallel, and an angle A'BB' 
being made equal to the latitude of that parallel, we shall have the dif- 
ference of longitude, represented by the hypotenuse A'B. We thus 
have the following theorems, viz. in the triangle A''B'B, 

COS. A'BB' : BB' : : radius : BA' ; 
that is, I. Cos. mid. lat. : departure : : radius : diff. of long. 

In the triangle A'BA, sin. A' : AB : : sin. A : A'B ; 
that is, n. Cos. mid. lat. : distance : : sin. course : diff. long. 

In the triangles ABB', A'BB', 

AB' tan. A = B'B ; A'B cos. A'BB' = B'B; therefore, 

AB' : A'B : : cos. A'BB' : tan. A ; that is, 
m. Diff. lat. : diff. long. : : cos. mid. lat. : tan. course. 

These three proportions comprise the theory of middle latitude sail- 
ing, and when to the middle latitude the proper correction, taken from 
Mr. Workman's table is added, these theorems will be rendered strictly 
accurate. 

EXAMPLES. 

1. A ship, in latitude 51° 18' N., longitude 22° 6' W., is bound to a 
place in the S.E. quarter, 1024 miles distant, and in lat 37° N. : what 
is her direct course and distance, as also the difference of longitude 
between the two places. 

Lat. from 51° 18' N. > o c ^ .-. j ooo 10/ 

Lat to 37' N f Sum of latitudes 88° 18' 

' Mid. lat. 44 9 

Diff. lat. 14 18 = 858 miles. 



For the course. 
As distance 1024 30103000 
: radius 10- 

: : diff. lat. 858 2-9334873 



: COS. course 33° 5' 9-9231873 



For the diff. long, 
cos. mid. lat. 44° 9' ar. com. 
: tan. course 33 5 
: : diff. lat. 858 



diff. long. 



79 



01441668 

9-8138993 
2-9334873 

2-8915534 



76 PLANE AND SPHERICAL TRIGONOMETRY. 

In this operation the middle latitude has not been corrected, so that 
the difference of longitude here determined is not without error. To 
find the proper correction look for the given middle latitude, viz. 44" 
9' in the table of corrections, the nearest to which we find to be 45° ; 
against this and under 14° diff. of lat. we find 27', also under 15°, we 
find 31', the difference between the two being 4'; hence corresponding 
to 14° 18' the correction will be about 28'. Hence the corrected middle 
latitude is 44° 37', therefore, 

COS. corrected mid. lat. 44° 37' ar. comp. 0.1483780 

: tan. course 33 5 9-8138993 

: : diff. lat. 858 2-9334873 



: diff. long. 786-6 28957646 ; 

therefore, the error in the former result is about 7i miles. 

2. A ship sails in the N. W. quarter, 248 miles, till her departure is 
135 miles, and her difference of longitude 310 miles : required her 
course, the latitude left, and the latitude come to. 

Course N. 32° 59' W. ; lat. left 62° 27' N. ; lat. in 65° 55' N. 

3. A ship, from latitude 37° N., longitude 9° 2' W., having sailed 
between the N. and W., 1027 miles, reckons that she has made 564 
miles of departure ; what was her direct course, and the latitude and 
longitude reached'? 

'Course N. 33° 19' W. or N. W. by N. nearly : lat. 51° 18' N. • long 22° 
8'W. 

4. Required the course and distance from the east point of St. 
Michael's, lat. 37° 48' N., long. 25° 13' W., to the Start Point, lat. 50° 
13' N., long. 3° 38'j the middle latitude being corrected by Workman's 
Table. Course N. 51° 11' E.; distance 1189 miles. 

Mercator's Sailing. 

{68.) It has been already seen that when a ship sails on any oblique 

rhumb, the difference of latitude, the departure, and the (V q 

/distance run, are truly represented by the sides of a 
right-angled plane triangle. The departure B'B repre-^' 
sents the sum of all the very small meridian distances, 
or elementary departures, 6'Z>, c'c, &c. in the diagram, at 
page 74, the difference of latitude AB represents the sum / 
of all the corresponding small difference in the figure ^, 
referred to ; and the distance AB, the sum of all the dis- 
tances to which these several departures and differences -^ 
belong, and each of these elements are supposed to be taken so exces- 
sively small as to form on the sphere a series of triangles, differing 
insensibly from plane triangles. 

Let Ab'b in the annexed diagram represent one of these elementary 
triangles, b'b will be one of the elements of the departure, and Ab', the 
corresponding difference of latitude ; and as b'b is a small portion of a 
parallel of latitude, it will be to a similar portion of the equator, or ot 
the meridian, as the cosine of its latitude to radius (66). This similar 
portion of the equator, or of the meridian, being the difference of longi- 
tude between b' and h. Suppose now the distance A^ prolonged to p, 
till the departure p'p is equal to the difference of longitude of Z)', and b, 
then b'b will be to p'p as the cosine of the latitude of b'b to the radius ; 
but b'b -.p'p :: Ab' : Ap' ; hence the proper difference of latitude Ab' is to 
the increased difference Ap' as the cosine of the latitude of b'b to the 
radius. Calling, therefore, the proper difference of latitude d, the 
increased difference D, the latitude of b'b, C, and the radius R, we 

have D = = Rd see. I ; the ship, therefore, having made the small 

cos. I > r> 




PRINCIPLES OF NAVIGATION. 77 

departure b'b, and the difference of latitude Ab\ must continue her 
course till the difference of latitude becomes D, in order that her de- 
parture may become equal to the difference of longitude corresponding 
to b'b. Conceiving all the elementary distances to be in this manner 
increased, the sum of all the corresponding increased departures will 
necessarily be the whole difference of longitude made by the ship during 
the course ; to represent, therefore, the difference of longitude due to 
the departure B'B, and difference of latitude AB', we must prolong AB' 
tni AC is equal to the sum of all the elementary differences increased 
as above, and the departure C'C, due to this difference of latitude, will 
represent the difference of longitude actually made in sailing from A 
to B. The determination of AC requires the previous determina- 
tion of all its elementary parts ; if (i be taken equal to 1', each of 
these parts will be expressed by D = 1' sec. Z, from which equation the 
values of D, corresponding to every minute of I, from the equator to the 
pole, may be calculated ; and by the continued addition of these there 
will be obtained, in succession, the values of the increased latitude cor- 
responding to r,2', 3', &c. of proper latitude ; these values are called the 
meridional parts^ corresponding to the several proper latitudes, and 
when registered in a table, form a table of meridional parts, given in all 
books on Navigation. 

The following may serve as a specimen of the manner in which such 
a table may be constructed, and, indeed, of the manner in which the first 
table of meridional parts was actually formed by Mr. Wright^ the pro- 
poser of this ingenious and valuable method. 
Mer, pts. of 1' = nat. sec. 1'. 
Mer. pts. of 2' = nat. sec. 1' -j- nat. sec. 2'. 
Mer. pts. of 3' = nat. sec. l'-{-iiat.sec.2'-f-nat. sec. 3'. 
Mer. pts. of 4' = nat. sec. 1' -j- nat. sec. 2'-]- nat. sec. 3' -j- nat. sec. 4'. 

&c. &c. 

Hence, by means of a table of natural secants, we have 

Nat. sees. Mer. parts. 
Mer. pts. of r = . . TOOOOOOO = 1-0000000 
Mer. pts. of 2' = 1-0000000 + 1 0000002 ^ 20000002 
Mer. pts. of 3' = 20000002+ 1-0000004 = 30000006 
Mer. pts. of 4' = 30000006 + 1-0000007 = 4-0000013 
&c. &c. 

There are other methods of construction, but this is the most simple 
and obvious ; we shall, however, presently have to advert to another pro- 
cess of computation, by which the meridional parts for any latitude may 
be found independently of previous calculations. The meridional parts, 
thus determined, are all expressed in geographical miles, because in the 
general expression D ~ Tsec. Z, V is a geographical mile. 

Having thus formed a table of meridional parts, (see Riddle's Naviga- 
tion, or Robertson's Treatise,) if we enter it with the latitudes sailed from, 
and come to, and take the difference of the corresponding parts in the ta- 
ble, the remainder will be the meridional difference of latitude, or the 
line AC in the preceding diagram, and the difference of longitude CC 
will then be obtained by this proporticm, viz. 

1. As radius is to the tangent of the course, so is the meridional differ- 
ence of latitude to the difference of longitude ; or if the departure be given 
instead of the course then the proportion will be 

2. As the proper difference of latitude is to the departure, so is the meri- 
dional difference of latitude to' the tangent of the course. Other proportions 
immediately suggest themselves from the preceding figure. 

(69.) As an example of Mercator's, or more properly of Wright's, sail- 
ing, let us take the following. 
7* 



78 PLANE AND SPHERICAL TRIGONOMETRY. 

1, Required the course and distance from the east point of St. Michael's 
to the Start Point. 

Start la.t.50°13'N. Mer.pts. 3494-8 long. S^SS'W. 

St. Michael's lat. 37 48 N. Mer.pts. 2453-1 long. 25 13 W. 



12 25 Mer.diff. lat. 0141-7 diff. long. 21 35 W. 
60 60 



Proper diff. lat. 745 miles. 1295 miles. 



For the course 
As Mer. diff. lat. 1041-7 3-0177427 
: radius , . 10- 
:: diff. long. 1295 3-1122698 



For the distance. 
As cos. course . 9-7971501 
: prop. diff. lat. . 2-8721563 

: : rad. . . 10- 



: tan. course 51° irE. 100945271 : distance 1189 30750062 
(70.) In the absence of a table of meridional pa.rts, a table of loga- 
rithmic tangents may be employed for the same purposes ; and, indeed, 
the meridional parts corresponding to any given latitude maybe expe- 
ditiously computed by help of such a table, and independently of any 
previous computations. 

It was shown, at page 76, that if a ship in latitude .r, vary tier lati- 
tude by a very small portion A.r, and that she continue her course 
till her departure equals the difference of longitude due to the difference 
of latitude A.^, then the enlarged difference of latitude (A?/), due to 

A7/ 
this departure, will be Ay — sec. x Ax /. — ^ = sec. x. This expression, 

it must be remembered, is nearer the truth the smaller we suppose Ax 
to be, and is, therefore, accurately true only when A.r = : in other 

Ay 
words, sec. x is the value to which the ratio —^ continually approaches, 

as we continually diminish Ax, (and in consequence Ay,) and which 
value it actually becomes only when the terms of the ratio vanish, and 

the fractions take the form — - By adopting the language of the Dif- 







dy 



Caladus we have, in this case, -r- = sec. x .'. dij = sec. xdx = 
dx "-^ 

— '- — .•. y — log. tan. (45° -\~^x), see Int. Calculus, p. 69 ; the logarithm 

here used is the Naperian. To change it into a common logarithm we 
must multiply by the modulus 2-302585, &c. ; it must be observed, however, 
that it is the logarithm of the natural tangent which is here expressed, and 
not the tabular logarithmic tangent; it is, therefore, equal to the tabular 
logarithmic tangent minus 10. Hence, emplojdng the table of logarith- 
mic tangents, we maj^ compute y from the formula y = 2-302585 Uog. 
tan. (45° -f- J x) — 10 ( X Rad. and thus, as stated above, the meridional 
parts, y, corresponding to any given latitude x, may be expeditiously 
computed, independently of any previous computations. 

The tables of meridional parts are usually expressed in nautical 
miles, and we shall have the number of miles in y, if, instead of multi- 
plying by the radius of the earth, we multiply by the number of miles or 
minutes in it. Now in every circle the radius is equal to 34-3774679 mi- 
nutes of that circle, because 3-14159, &c, : 180° :: 1 : 343-774679 minutes; 
hence for the number of miles in y the expression is 7915-7044679 
jlog. tan. (45° -|- | 2:) — lOj; or, since tan. 45° + J a; = cot. 45° — fa;, 

and, since, moreover, log. cot. = log. = 20 — tan. ; this expression 

tan. 



PRINCIPLES OF NAVIGATION. 79 

may be written thus, 7915-7044679 J 10 — log. tan. (45° — I a;) J , which 
gives the rule in the text. We had intended to have introduced here 
some other particulars relating to Mr. Wright's projection of the meri- 
dian line, but we are precluded from doing so, as this treatise has already 
exceeded the limits assigned to it. We must, therefore, content our- 
selves with referring the student to Robertson's Nav. vol. n. p. 135 — 146. 

The practical rule is as follows, viz. if the log. tangent of half the com- 
plement of any latitude be subtracted from 10, and the remainder be mul- 
tiplied by 79157044679, &c. the product will give the meridional parts 
in miles, corresponding to that latitude. 

From this rule the method of operating with logarithmic tangents, in- 
stead of with meridional parts, may be easily derived. Call t, t', the loga- 
rithmic tangents of the half complements of the latitudes left and reached, 
and put a for the constant multiplier 7915-7044, &c. Then, by the rule 
just given, the meridional difference of latitude will be 

10000 
aiao — n — {10 — t)l= a {t—t')=^(t— t') 10000 -f- • 

Now log. = -1015104, therefore, the logarithm of the meri- 
dional difference of latitude is found by removing the decimal point in 
the difference t — t' four places to the right, and then subtracting the 
constant num.ber -1015103. Hence, if instead of the logarithm of the 
radius 10, we use 10-1015104, and instead of the meridional parts the 
logarithmic tangents t, t', of the complements of the half latitudes, taking 
care in setting down the difference of these to remove the decimal point 
four places to the right, the proportion (1), at page 77, may be still em- 
ployed. Thus, taking the foregoing example, the operation by this 
method will be as follows. 

. ^ 1 , 5 St. Michael's 26° 6^ .-. t = 9-6901030 
a 1^0. lat. ^ g^^^^ 19 53J .-.r^ 9-5585051 

t—t'^ 1315-979 



Ast — f 1315-979 arith. comp. - 6-8807448 
: Const, log. - ... - 10- 1015104 
::diff. long. 1295 - - - 31122698 

: tan. course N. 51° IFE. - - 100945250 

The reason why the resulting logarithm here does not exactly coin- 
cide with that obtained by using the meridional parts, is that the meri- 
dional parts have been computed to but one place of decimals ; if they 
had been computed to two or three places, the two results would have 
been exactly the same. 

2. Given the Lizard in lat. 49° 55' N. Barbadoes in lat. 13° 10' N. and 
their difference of longitude 53°, or 3180' W. ; to determine the course 
and distance. Course S. 49° 59' W. ; distance 3429 miles. 

3. A ship sails from lat. 37° N. long. 22° 56' W., on the course N., 
33° 19' E., till she arrives at lat. 51° 18' N. : required the distance sailed, 
and the longitude arrived at. 

Distance 1027 miles; longitude in 9° 45' W. 
We shall here terminate the present chapter on the principles of Navi- 
gation, having now discussed the several cases of s'ailing which actually 
occur in practice. But the student who is desirous of prosecuting his 
inquiries on this very important branch of practical science to greater 
extent, will, of course, consult works expressly devoted to the subject. 
Of these, the most elaborate in our language is the valuable " Elements" 
of Robertson, in two octavo volumes. The Treatise of Mr. Riddle is 
also an excellent work, abounding with practical examples very accu- 



80 PLANE AND SPHERICAL TKIGONOMETRT. 

ratelj solved, and upon the whole, better adapted to modern practice, as 
well as more compendious, than Robertson's. Mr. Norie's Navigation 
is also a good practical book, and so is that of Dr. Bowditch. 



CHAPTER 11. 

APPLICATION OP SPHERICAL TRIGONOMETRY TO ASTRONOMICAL PROBLEMS. 

(71.) The solution of Astronomical Problems forms one of the most 
useful and agreeable applications of the theory of spherical Trigono^ 
metry. To such inquiries the theory itself, no doubt, owes its origin, as 
well as many of the successive improvements which it has gradually 
received, so that a specimen of its use in the solution of astronomical 
problems may reasonably be looked for in a book on Trigonometry. 

For the purpose of measuring the angular distances of the heavenly 
bodies from each other, and from the horizon, it is convenient to suppose 
them all situated as they really appear to an observer on the earth, viz. 
in a spherical concave surrounding our earth and concentric with it.- 
This imaginary concave is called the celestial sphere, or the apparent 
heavens ; in it all the apparent motions of the heavenly bodies are, for 
the convenience of trigonometrical application, supposed actually to 
take place ; and the entire celestial sphere to revolve daily round the 
earth, as if this were at rest in its centre. All this is allowable, because 
the applications of which we speak are not affected b}'- the inquiry, 
whether the motions which the heavenly bodies present to an observer 
on the earth are really as they appear or not. 

At the opening of last chapter we defined several lines which geogra- 
phers had found it convenient to consider as described on the surface 
of the earth ; most of these, astronomers extend to the heavens. Thus 
the plane of the earth's equator, when extended to the heavens, marks 
on the celestial sphere the great circle called the equinoctial, and in like 
manner, the meridians being extended to the heavens, mark out the 
celestial meridians ; also the axis of the earth, about which its real mo- 
tion takes place, when extended to the heavens, is the axis about which 
the apparent motion of the celestial sphere takes place : this axis marks 
out the north and south poles of the heavens. 

As the sun performs its apparent revolution about the earth in 24 
hours it passes over \b° in an hour ; if then we consider, as astronomers 
do, that the day at any place commences at noon, or when the sun is on 
the meridian of that place, the time shown by the sun in any position 
will be expressed in degrees by the arc of the equinoctial, intercepted 
between the fixed meridian of the place, and that passing through the 
sun, or it will be expressed by the angle included by these meridians. 
Celestial meridians are, therefore, also called hour circles^ and the angle 
between the meridian of the place and that through the sun is called the 
hour angle, or the horary angle. That meridian which is at right-angles 
to the meridian of the place is the six o'clock hour circle, since the sun 
obviously reaches it when halfway between noon and midnight. 

Besides these lines, thus transferred from the earth to the heavens, 
there are others peculiar to the celestial sphere, which must be men- 
tioned ; these are, 1st, ecliptic, which is the great circle path described 
by the sun among the fixed stars in its apparent annual motion about the 
earth : in reality it is the path of the earth moving in a contrary direc- 
tion about the sun. This circle crosses the equinoctial at an angle sub- 
ject to an exceedingly small variation, determinable by observation and 
computation ; its inclination to the equinoctial is about 23° 28', but it is 
always given with the minutest attamable accuracy in the Nautical 



ASTKONOMICAL PROBLEMS 81 

AVnianack. The points where the ecliptic crosses the equinoctial are 
c^Si^Ai^xA equinoctial ]}oi%ts : the sun enters these points about the 21st 
of March and the 23d of September ; the former being called the vernal 
equinox, and the latter the autumnal equinox. These names are given 
because at such times the nights are equal in length to the days all over 
the world ; for as the two poles of the earth are at these times symmetri- 
cally situated with respect to the sun, the circular boundary, which 
separates the enlightened hemisphere from the darkened, must pass 
through both poles ; and hence any point on the earth will be as long in 
being carried, by the earth's uniform rotation, through the enlightened 
part as through the dark part. The meridian through the equinoctial 
points is called the equinoctial colure. 

The position of any point on the celestial sphere, like the position of 
a point on the terrestrial sphere, is marked out by its latitude and lon- 
gitude. On the celestial sphere the circle of longitude is the ecliptic; 
and perpendiculars to this, passing, therefore, through the poles of the 
ecliptic, are the circles of celestial latitude ; the point from which lon- 
gitude is measured is the vernal equinoctial point. Commencing at this 
point, too, the ecliptic is divided into twelve parts, called signs ; a sign 
is therefore 30°. The twelve signs are named, and symbolically ex- 
pressed, as follow : 
1.^ Aries. 4. o Cancer. 7.=^ Libra. 10. V5 Capricornus. 

2. ^ Taurus. 5. ^^ Leo. 8. m Scorpio. 11. ^ Aquarius. 

3. n Gemini. ) 6. n^ Virgo. 9. J Sagittarius. 12. >€ Pisces. 
The first six of these signs are on the north of the equinoctial, the 

others on the south, and the vernal equinoctial point is called the first 
point of Aries. The longitude is measured from this point in but one 
direction, viz. in the order of the signs. 

Besides the above method of marking out the position of a celestial 
body, by means of its latitude and longitude, there is another way, viz. 
by means of its Right Ascension and Declination. The right ascension is 
measured on the equinoctial from the first of Aries, in the order of the 
signs, and the declination is measured on the perpendicular to this, or 
circle of declination passing through the object. We see, therefore, that 
what on the terrestrial sphere is latitude a.nd longitude, is on the celestial 
sphere declination and right ascension ; and parallels of latitude on the 
one correspond to parallels of declination on the other. Of these the 
two which are 23° 28' from the equinoctial, one on each side, and which 
therefore touch the ecliptic in the first points of Cancer and Capricorn, 
are called the tropics of Cancer and of Capricorn. These first points 
of Cancer and Capricorn are respectively called the summer and winter 
solstice ; because for a day or two before and after the sun enters them 
he appears to be stationary, and the days to be of equal length, so 
slowly does his declination at those times change, for his motion is 
obviously very nearly parallel to the equinoctial. The meridian, through 
the solstitial points, is called the solstitial cohere, and that through the 
equinoctial points, the equinoctial colure. 

Having described the principal circles and points of the celestial 
sphere which are considered as permanent, or which do not alter with 
the situation of the observer on the earth, we come now to describe 
those which change with his place. The principal of these is the horizon^ 
which has been defined already (63), and vertical circles which are per- 
pendicular to the horizon, and on which the altitudes of celestial objects 
are measured. 

These vertical circles all meet in two points diametrically opposite, 
viz., the poles of the horizon ; that one which is directly over Ihe head 
of the observer is called his zenith, and the opposite "one hii^ nadir. 
That vertical which passes through the east and west points of the liori- 




82 PLANE AND SPHERICAL TRIGONOMETRY. 

zon is called the prime vertical ; it necessarily intersects the naeridian 
of the place (which passes through the north and south points) at right- 
angles. 

The azimuth of a celestial object is an arc of the horizon, comprised 
between the meridian of the observer and the vertical circle through the 
object, and hence vertical circles are sometimes called azimuth circles. 

The amplitude of a celestial object is the arc of the horizon, comprised 
between the east point and the point where the object rises, or between 
the west point and that where it sets; the one is called the rising am- 
plitude, the other the setting amplitude. These definitions and remarks 
will suffice to render the following problems intelligible. 

PROBLEM I, 

(72.) Given the sun's right ascension and declination to determine his 
longitude and the obliquity of the ecliptic. 

Let n EsGl represent the celestial meridian through the first of Cancer 
and Capricorn, that is, let it be the solstitial colure, ns the axis of the 
sphere, EGl the equator, eC the ecliptic, and %Ss the 
declination circle, passing through the sun S; then y 
ARS is a right angle, and in the right-angled spherical e^ 
triangle ARS there are given the right ascension AR, 
and the declination RS to find the longitude AS, and 
the obliquity SAR, which is an easy operation in right- 
angled spherics. It is necessary, however, to remark 
that as celestial longitude and right ascension are measured from A, the 
first point of Aries in the direction AS of the signs quite round the celes- 
tial sphere, when, of the four quantities in the problem, the obliquity 
and the declination are given to find the others, we must know on what 
side of the equinoxial the sun is, that is, whether the declination is north 
or south, for if the sun have the north declination RS, the longitude will 
be AS ; but if it have the equal south declination R'S', the longitude 
being measured in the direction ASC round the globe to S', will be, 
instead of A'S', 360°— A'S'. 

It is moreover necessary to know not only on which side of the equi- 
noctial the sun is, but also on which side of the tropic ; for the sun, in 
passing from a tropic to the equinox, descends through the same gra- 
dations of declination as it ascended through in passing from the pre- 
ceding equinox to the tropic, although its longitude and right ascen- 
sion goes on increasing; in addition, therefore, to knowing whether the 
declination is north or south, we must also know whether it be increasing 
or decreasing, in order to determine the longitude and right ascension 
without ambiguity ; and these particulars will be known from knowing 
the time of the year when the proposed declination is observed ; thus 
from the 21st of March to the 21st June, during which time the sun is 
in the first quadrant of the ecliptic, the sun's declination is north and 
increasing ; it afterwards continues to decrease, still remaining north, 
during the second quadrant, that is till the 23d of September, from which, 
till the 21st of December, that is, during the third quadrant, the declina- 
tion is south and increasing, after v/hich or during the fourth quadrant, 
the declination still south, decreases till the 21st of March, 

EXAMPLES. 

1. Given the sun's right ascension on the 17th of May, 53° 38', and 
its declination 19° 15' 57" ; required his longitude and the obliquity of 
the ecliptic. 

Applying Napier's rule to the right-angled triangle ARS, we have 

Rad. X cos. AS = cos. AR cos. RS. 

■r, 1 * T^ T^o. . » Rad. sin. AR 

Rad. sm. AR = tan. RS cot. A .-. cot. A = ^r— — . 

tan. RS 



ASTRONOMICAL PROBLEMS. ' 83 



Hence the computation for AS and A is as follows. 



For the longitude AS. 
COS. AR 53° 38' 0" 9-7730185 
COS. RS 19 15 57 99749710 



For the obliquity A. 
sin. AR 9-9059247 

tan. RS arith. comp. 0-4565209 

COS. AS 55 57 43 9-7479895 cot. A 23° 27' 50i" 10-3G24456 

On the 31st of March, 1816, the sun's declination was observed at 
Greenwich to be 4° 13' 3U" : required his right ascension, the obliquity 
of the ecliptic being 23° 27' 51". The right ascension was 9° 47' 59". 

3. Required the sun's longitude on the 28th of November, 1810, when 
his declination was 21° 16' 4", and his right ascension, in time, 16'' 14'" 
58-4% or in degrees 243° 44' 36". 

The longitude was 245° 39' 10", or 8 signs 5° 39' 10." 

4. The sun's longitude being 8^^ 7° 40' 56", and the obliquity of the 
ecliptic 23° 27' 42i" : required his right ascension in time. 

The right ascension is IG'' 23'" 34'. 

PROBLEM n. 

Giving the sun's declination to find the time of his rising and setting 
at any place whose latitude is known. 

Let nEsQi represent the meridian of the place, Z 
being the zenith, and HO the horizon, and let s' s" 
be the apparent path of the sun on the proposed day, 
cutting the horizon in S. Then the arc EZ will be 
the latitude of the place, and consequently EH, or its 
equal 0,0, will be the colatitude, and this measures 
the angle O AQ, ; also RS will be the sun's declination, 
and AR, expressed in time, will express the time of 
sunrise from 6 o'clock, for nAs is the 6 o'clock hour circle. 

Hence, in the right-angled triangle ARS, we have given RS and the 
opposite angle A to find AR, the time from 6 o'clock. 

EXAMPLE I. 

Required the time of sunrise at latitude 52° 13' N., when the sun's 
declination is 23° 28'. 

By Napier's rule, Rad. sin. AR = cot. A tan. RS = tan. lat. tan. dec. 
tan. 23° 28' - - 9-6376106 

tan. 52 13 - - 10- 1105786 




sin. 34 3 2U" - 9-7481892 
4* 



AR in time 2'' 16' 13" 25' 
6 



3 43 46 35 == time of rising. 



SCHOLIUM. 



It should be here remarked that the time thus determined is apparcniy 
time, which is that which would be shown by a clock so adjusted as to 
pass over 24 hours during one apparent revolution of the sun, or from 
its leaving the meridian to its return to it again, the index pointing 
to 12, when the sun is on the meridian. But it is impossible that any 
clock can be so adjusted, because the interval between the successive 
return of the sun to the meridian is continually varying, on account 
of the unequal motion of the sun in its orbit, and of the obliquity 
of the ecliptic ; each of these varying intervals is called a true solar 

' Degrees are converted into hours by inultiplying by 4 and dividins; by 60. 



84 PLANE AJST3 SPHERICAL TRIGONOMETRY. 

day, and it is the mean of these during the year which is measured 
by the 24 hours of a well regulated clock, this period of time being 
a mean solar da}-; hence, at certain periods of the )'ear, the sun will 
arrire at the meridian before the clock points to 12, and at other 
periods the clock will precede the sun ; the small interval between the 
arrival of the index of the clock to 12 and of the sun to the meridian is 
called the equation of time, and it is given in page ii. of the Nautical 
Almanack for ever\' day in the year ; this correction, therefore, must 
always be applied to the apparent time determined by trigonometrical 
calculation to obtain the true time, or thai shuw n by a well regulated 
clock or chronometer. 

Another circumstance too must be taken into account, in order to 
determine the apparent time with rigorous accuracy, viz. the change in 
the declination of the sun from sunrise to noon. In the Nautical Al- 
manack the declination of the sun is given for every day at noon, and if 
this be used in the computation, we shall assume that the declination 
has not varied from sunrise to noon, which is not the case ; hence it will 
be necessary to compute the declination for the time of sunrise, as de- 
termined above, and then to resolve the problem with this corrected de- 
clination. The correction is obtained by taking from the Nautical 
Almanack the variation of declination in 24 hours, and then finding by 
proportion the variation for the time required. 

2. Required the time of sunrise at latitude 57° 2' 54'', when the sun's 
declination is 23^ 28' % 3^ ll'^ 49^ 

3. How long is the sun above the horizon in latitude 58^ 12' N., when 
his declination is 18° 40' S. 1 1^ 35^" 52^ 

proble:m m. 

Given the latitude of the place, and the declination of a heavenly 
body, to determine its altitude and azimuth when on the six o'clock 
hour circle. 

Let HZPO be the meridian of the place, Z the 
zenith, HO the horizon, S the place of the object on 
the six o'clock hour circle PS^, which of course 
passes through the east and west points of the hori- 
zon, and ZSB the vertical circle passing through 
the sim. Then in the right-angled triangle SBA, 
the given quantities are AS. the declination, and 
the arc OP, or angle SAB, the latitude of the place, 
to find the altitude BS, and the azimuth BO from the north point O of 
the horizon ; or to find the complement AB of this azimuth, that is, the 
sun's bearing from the east. 

EXAMPLES. 

1. What was the altitnde and azimuth of Arcrarus, when upon the 
six o'clock hour circle of Greenwich, lat. 51° 28' 40" N., on the 1st of 
April, 1822 ; its declination on that day being 20° 6' 50" N. 1 

By Napier's rule we have Rad. sin. BS = sin. A sin. AS. 

Rad.cos.A=tan.AB cot. AS .-.cotBO^r .^^ . 

cot. AS 

For the altitude For the azimuth. 

sin. A 51° 28' 40" 98934103 | cos. A - - 9-7943612 

sin. AS 20 6 50 9-5364162 | cot. AS - 104362545 




sin. BS 15 36 27 94298265 | cot. BO 77° 9' 4" 9-3581067 

Hence the altitude is 15° 3G' 27", and the azimuth 77° 9' 4" N. . 



ASTRONOMICAL PROBLEMS. 



2. At latitude 62° 12' N. the altitude of the sun at 6 o'clock in the 
morning was found to be 18° 20' 23" ; required his declination and 
azimuth. Declination 20° 50' 12" N., Azimuth 79° 56' 11" from N. 

3. On the 20th of November, 1822, the declination of Aldebaran was 
16° 8' 36" N., what was its altitude and azimuth when on the six o'clock 
hour circle of Greenwich, lat. 51° 28' 40" N. '? 

Altitude 12° 32' 3", Azimuth 79° 46' 50" from N. 

PROBLEM IV. 

The latitude of the place and the declination of the sun being given 
to find the time when it is due east, or upon the prime vertical, and the 
altitude at that time. 

Having drawn the meridian of the plane as be- 
fore, the vertical circle ZAN, at right-angles to it, 
will be the prime vertical, A being the east point of 
the horizon HAO : also P being the elevated pole, 
and S the place of the sun, ZP will be the colatitude, 
PS the codeclination, ZS the coaltitude, and ZPS 
the hour angle, or time from noon; hence, in the 
right-angled spherical triangle SZP, there are given 
SP and PZ, to find SZ and the angle P. If the declination ''is not of 
the same name as the latitude, the sun will arrive at the prime vertical 
at S' before it rises : in this case the declination is to be considered as 
negative. 

By Napier's rule Rad. cos. P = cot. SP tan. PZ = tan. dec. cot. lat. 

-, J „_ „ry -ofT- • 1, Rad. sin. dec. 

Rad. COS. SP = cos. SZ cos. PZ .-. sm, alt. = : — ^ 

sm. lat. 




:examples. 



I. On the 1st of August, 1831, the sun's declination was 18° 10' 22" 
N., at what hour was he due east at Greenwich, and what was his alti- 
tude at that time'? 



For the hour angle. 
tan. dec. 18° 10' 22" 95162138 

cot. lat. 51 28 40 9-9009509 



cos. hor. angle 74 51 7 
4 



For the altitude, 
sin. dec, 
sin. lat. 



9-4939924 
9-8934103 



9-4171647 sin. alt. 23° 29' 37" 9-6005821 



4''59'"24^28« 
12 



7 35 32. Hence the time is 35 seconds and a half past 
7 o'clock, and the altitude 23° 29' 37". 

2. Given the sun's declination 5° 8' 26" N., and his altitude when due 
east 16° 53' 10" ; required the latitude of the place. 

Latitude 17° 58' N. 

3. If the declination of a celestial object be 18° 4' S., what is its alti- 
tude when on the prime vertical of latitude 27° 42' S., and its distance 
from the meridian in time 1 

Altitude 41° 51' : Merid. distance in time 3^" 26'" 20^ 



PROBLEM V. 



To find the time when the apparent motion of a celestial object is 
perpendicular to the horizon, from having its declination and the lati- 
tude of the place given. 



PLANE AND SPHERICAL TRIGONOMETRY. 



Let 5 s' represent the parallel of declination, or the 
apparent diurnal path of the body, and let the verti- 
cal ZSN be drawn to touch it in S ; then S will be the 
place of the body when its apparent motion is in the 
direction SZ of the vertical, and therefore perpendi- 
cular to the horizon : through S draw the hour circle 
PS, which being the shortest distance from P to ZN, 
is perpendicular to it (p. 53) ; hence the triangle 
PSZ is right-angled at S, and in which we have 
given the colatitude ZP, and the codeclination PS, to find the hour 
angle ZPS. 

It is obvious that this problem will be impossible when Ps' exceeds 
PZ ; that is when the declination is less than the latitude. 





PROBLEM VI. 

To determine upon what vertical a celestial object must be, in order 
that a small error, committed in taking its altitude, may have the least 
possible efiect upon the hour angle. 

Let S be the place of the sun or other body, but by 
an error in taking its altitude let it be referred to S'. 
Draw S' S" parallel to the horizon, and meeting the 
parallel of declination s 5' in S", then when the body 
is at S'^ it will really have the coaltitude ZS" = ZS', 
which it M^as erroneously supposed to have at S, so that 
in the determination of the hour angle P, from the 
colatitude, the coaltitude, and the codeclination, the 
small angle S' PS" will be the amount of the error. 

As the triangle SS' S'' is, of course, exceedingly small, it may be 
regarded as a rectilinear triangle, right-angled at S' ; therefore SS' = 
SS'' sin. S" and SS" = sin. PS. < SPS'', (see page 73,) consequently, 

SS' = sin. S'' sin. PS • < SPS" .-. SPS''r= -. — ^,^^.' ^^ . (1). 

sm. S ' sm. PS ^ ^ 

Now the angle S" is equal to the angle ZSP, because S' SS'' is the 
complement of each, and therefore, by the relation between the sides 
and angles of a spherical triangle, we have 

sin. S" : sinPZ :: sin. SZP : sin. PS 
.-. sin. S'' sin. PS =z sin. PZ sin. SZP. 
Substituting the second member of this equation in (1), we have 
SS' _ error in alt. 

~ sin. PZ sin. SZP ~ cos. lat. sin. azimuth' 
This expression will obviously be the least possible when the sine of 

the azimuth is the greatest possible, or when the azimuth is 90° ; that 

is, when the body is on the prime vertical. 

Hence, indeducing the time from an altitude of any celestial body, 

it will be best to make the observation when the body is either exactly, 

or nearly due east or due west. 



PROBLEM VIL 



The latitudes and longitudes of two celestial objects being given to 
determine their distance apart. 

Let P represent the pole of the ecliptic, and PS, ^ 

PS', two arcs of celestial latitude, drawn to the two 
objects S S' ; then will these arcs represent the colati- 
tudes, the angle P will be the difference of longitude, 
and the arc SS' will be the distance sought, so that 
we have two sides and their included angle given to 
find the third side. In order to this we must first de- §■ 




ASTRONOMICAL PROBLEMS. 



S7 



termine agreeably to the method explained at page 60, a subsidiaiy 
angle, w, by the equation cot. w = tan. PS cos. P ; after which the side 
^o . . .. . • oo, cos.PSsin.(a)+PS') 

SS' IS found by the equation cos, SS' = -. 

•' sin. cj 



EXAMPLES. 



1. Required the distance between Procyon and Capella, the latitude 
of Procyon being 15° 58' 14'' S., and its longitude 3^' 22° 55' 42" ; also the 
latitude of Capella being 22° 51' 57" N., and longitude 2^ 18° 57' 57"'? 

Taking the difference of the longitudes, we have for the angle P, 
P = 33° 57' 45" ; and for the polar distances we have PS = 105° 58' 14", 
PS' = 67° 8' 3" ; hence the logarithmic process will be as follows : 
tan. PS 105° 58' 14" - 10-5433466 cos. PS 9-4395590 

cos. P 33 57 45 - 9-9187658 sin. w, ar. comp. 04865396 



cot. « 160 57 46 - 10-4621124 sin. (co + PS') 9-8717340 



PS' =67 8 3 COS. SS' 51° 6' 39" 9-7978326 



to + PS'=:228 5 49. 

In this example cot. m is negative, because tan. PS is negative, and 
COS. P positive ; also cos. SS' is positive, because cos. PS is negative, 
sin. (w-f-PS') negative, and sin. w positive. The operation will obvi- 
ously be similar, when, instead of the latitudes and longitudes, the right 
ascensions and declinations of the two bodies are given to find their dis- 
tance apart. 

2. The latitude and longitude of a star S. are 38° 40' 26" N., and 3s 
2° 4' 40' ; and of a star S'., 13° 26' 11" N., and 9' 11° 41' 26" ; required 
their angular distance apart. Distance 127° 7' 11". 

3. "What is the distance between Sirius and Procyon, the right ascen- 
sion of Sirius being 99° 0' 21", and its declination 16° 26' 35" S.; and 
the right ascension of Procyon 112° 6' 47", and its declination 5° 45' 
3" N. Distance 25° 42' 10". 

PROBLEM Vm. 

Given the latitude of the place and the sun's declination to find the 
beginning and end of twilight. 

Twilight commences in the morning and ends in the evening, when 
the sun is about 18° below the horizon. Hence, if PZ (see the diagram 
to next problem) represent that portion of the meridian which is inter- 
cepted between the elevated pole and the zenith, and S' be that point in 
the sun's apparent path on any day which is 108° from Z, S' Av^ill be the 
place of the sun at the commencement of morning twilight, or at the 
termination of evening twilight ; also PS' will be the codeclination, and 
PZ the colatitude ; we thus have the three sides of the triangle PS'Z, to 
find the angle P. Hence, calling the sum of the three sides S, the for- 
mula for computing the hour angle P will be 

. ^_, I sin.(|S — ZP)sin.aS — PS') ^. ^ . ^ 

sm. I P = ^ -. — ^^^~. — Vr; ' : which is the same as 

N sm. ZP sm. PS' 

. ^ I sin, h (lat. H- 18° + codec.) sin, h (dec. -\- 18° -j- colat.) ^ 

'^ cos. lat. COS. dec ' 

a very convenient form for computation. 

EXAMPLES. 

1. At what time did tv/ilight commence at Edinburgh, lat. 55° 57' 
20" N., on the 20th of August, 1831, when the sim's declination was 12^ 
38' 9" N. 1 



88 



PLANE AXD SPHERICAL TEIGOX"0:'IETRY. 

lat.-{-18^ 73" 57' 20" 

codec. 77 21 51 ar, coinp. sm. 0-01064S0 





2)151 19 11 






75 39 35i sin. 


9-9862531 


dec.-f 
colat. 


-18^ 30 38 9 

34 2 40 ar. comp. 


sm. 0-2519393 




2) 64 40 49 
32 20 ^isin. 


9-72S3036 




2)19-9771490 




sin.iP 763 54'54r' • 9-9S85745. 

Hence P = 153^ 49' 49" = (in time) 10' 15^ 19i , so that twilight 
commeoced in the morning at I''- 44^" 40|', and ended in the evening at 
10^ 15 ■' 19i\ 

2. At what time does the twilight begin at latitude 48=' 38' 56" N., 
when the sun's decUnation is 8= 28' 54" ]S".l Twilight begins at 3'' 20'". 

3. At what time does twilight end at latitude 52= 12' Sa" IN"., when 
the sun's declination is 15= 55' 25" N.'? Twilight ends at lO'' 12^'". 

PROBLEM IX. 

Given the latitude of the place to determine on what day of the year 
the twilight is the shortest, and its duration on that day. 

Let HO represent the horizon, and ho the pa- 
rallel to it, 18= below it; also let PS be the de- 
clination circle, passing through the stm at 
sunset, and PZ, that passing through the zenith. 
Conceiving these two circles to revolve with S, 
PS will come to PS' when S comes to S'. andij^ 
PZ will take some determinate position PZ'. ' 
is'ow, since the angles ZPS. Z' PS', are equal. 
we have, by taking from each the common part Z'PS, ZPZ' = SPS'; 
but SPS', converted into time, expresses the duration of twilight, ZPZ' 
is therefore the least possible when the twilight is the shortest possible. 
]S'ow since the sides PZ, PZ'. are both given, the side ZZ' will be the 
shortest when the opposite angle, P, is the least ; (see equi. (A) p. 47.) 
hence when ZZ' is the shortest, the twilight is the shortest; but as the 
two sides Z'S', ZS, of the triangle ZZ'S', are given, the third side will 
be shortest when the angle S' is the least possible, and this is the case 
when Z' falls on ZS', for then the angle is 0. Hence the twilight is 
shortest when the angle PSZ is eqital to the angle PS'Z. 

Let then z be the proper position of Z' ; we shall have Z^ = ZA' — 
zh' — z'&' — z\' —h'S' — 18=, and because PZ =P~, the arc Fn, bisect- 
ing the angle ZFz. will also bisect the base Zz, and be perpendicular to 

• ^'--N , . „^ . ^^^ sin. Z?i sm. 9=' ,,^ 

it(£M:); consequentlv, sm. ZVn = sm.iSPS' = -^ — 7777- = -. — (l)j- 

co"? PZ sm.PZ COS. lat. ^ ''^ 

also COS. P?i = — ^" -; and, in the right-angled triangle P?iS', 

^.^°5. Z/t' ' C0S.7iS'C0S.PZ ^ . 

COS. Pb =cos. 7iS' cos.Fji= ^ : that is, 

COS. Zi7l 

sin. 99= sin. lat. sin. 9=^ sin. lat. „ . , 

sm. dec. = — -= — = — tan.9='om.lat. (*2). 

cos. 93 COS. 9=^ 



NAUTICAL ASTRONOMY. 89 

The declination bein^ known by this equation, the day of shortest 
twilight is also known, {Naut. Aim.) The declination will be of a con- 
trary name with the latitude as its sine is negative. Equation (1) ex- 
presses the duration of the twilight. Since the angles ZP^, SPS', are 
equal, the hour angles for the beginning and ending of the morning 
twilight, or for the ending and beginning of the evening twilight, are 
ZPS', .sPS'. Now, in the right-angled triangle P7iS^,we have 

„„, sin. S% sin. 99° cos. 9° ,„. 

sm. wPS' =^-. — -^ = — = — . . (3). 

sm. PS cos, dec. cos. dec. 

The sum of (1) and (3) gives the angles ZPS', and their difference 
the angle ;rPS' = ZPS, and thus we have the hour angles for the be- 
ginning and end of the twilight. 

EXAMPLES. 

1. Required the time and duration of the shortest twilight at Green- 
wich^lat 51° 28' 40", in the year 1832. 

For the duration, 
sin 9° . 9- 1943334 

cos. 51° 28' 40" 9-7943612 



For the day, 
tan. 9° . 91997125 

sin. 51° 28' 40" 98934103 



sin. dec. 7 7 5 9-0931228 



sin, 14 32 49 9-3999713 



The declination is therefore, 7° 7' 5" south, which {Naut. Aim.) cor- 
responds to March the 1st and to October 12. 

Also the hour angle SPS' is 29° 5' 38", which, in time, is 1'^ 56'" 22i«, 
the duration sought. 

To find the times of beginning and ending of the twilight, we have, 
from the equation (3) 

cos. 9° . 9-9946199 

cos. 7° r 5" . 9-9966399 



sin: 95 31 19 9-9979800. 

The angle tz-PS', thus determined, is obtuse, because its opposite side 
is greater than PS, and this is opposite to a right-angle. This angle, 
converted into time, is 6^ 22'" 5i^ Adding therefore, to this the angle 
ZPti, in time, that is half the duration, or 58"* lip, we have 7^^ 30^ 
16j*, the time when the evening twilight ends. Also, by subtract- 
ing the same quantity, we have 5'' 23'" 54"' for the time when the even- 
ing twilight commences. These results respectively taken from 13* 
leave the time when the morning twilight begins and ends. 



CHAPTER III. 

ON THE PRINCIPLES OF NAUTICAL ASTRONOMY, 

(73.) In our chapter on Navigation we have laid down several 
methods of determining the place of a ship at sea, by help of, the account 
kept on board of its progress through the water, that is, o*f the course 
and distance sailed ; and, if confidence could be placed in this account, 
even when kept with the utmost care, the art of Navigation would be 
perfect. Such perfection, however, it is hopeless to expect ; for it does 
not seem possible to measure, with strict accuracy, either a ship's rate 
or the direction in which she moves, both of which may indeed be con- 
tinually varying. In order, therefore, to determine the place of a ship 
at sea, with that accuracy which the safety of navigation requires, it is 
absolutely necessary that we be furnished with methods entirely inde- 
pendent of the dead reckoning^ and these methods it is the business of 
Nautical Astronomy to teach, 

8* L 



90 PLANE AND SPHERICAL TRIGONOMETRY. 

" It must not, however, be -understood that the dead reckoning is 
without its value ; on the contrary, when combined with astronomical 
observations, it is of considerable utility in detecting the existence and 
velocity of currents, and is iadispensa.bly necessary to fill up the short 
intervals which may occur in unfavourable weather between celestial 
observations. But the too general practice of relying exclusively upon 
it cannot be sufficiently deprecated, and numerous instances might be 
adduced of the fatal consequences of this reliance,- in the loss of vessels, 
from errors of such magnitude that they might have been detected by 
the most superficial knowledge of nautical astronomy, and the aid of 
even a good common watch." (^Capt. Kater's JNautical Astronomy in 
the Ency. Met.) 
On the Corrections to be applied to the observed Altitudes of Celestial Objects. 

(74.) The true altitude of a celestial object is always understood to 
mean its angular distance from the rational horizon of the observer. 
This is not obtained directly by observation ; but is the result of certain 
corrections applied to the observed altitude. These we shall now 
enumerate and explain. 

Of the Dip or the Depression of the Horizon. 

(75.) Let E represent the place of the observer's 
eye, and S the situation of any celestial body; the 
first object is to obtain its apparent altitude above 
the horizontal line EH; that is, the angular dis- 
tance SEH. Now, as to the observer, the visible 
horizon is EBH', the altitude given by the instru-/ 
ment is the angle SEH' ; hence we must subtract \ 
from this observed altitude the angle HEH', called 
the Dip or Depression of the Horizon, in order to 
obtain the apparent altitude SEH. 

The angle HEH', or its equal C, is calculated for various elevations, 
AE of the eye above the surface of the sea from the proportion, 

CE : EB = VeC-^— CB2 ; ; rad. : sin. C ; 
and the results are registered in a table. 

Of the Semidiameter. 

(76.) When the foregoing correction for dip has been applied, the re- 
sult will be the apparent altitude of the point observed above the hori- 
zontal plane through the observer's eye. If this pomt be the uppermost 
or lowermost point of the disc of the sun or moon, a further correction 
will be necessary to obtain the apparent altitude of the centre ; that is, 
we must apply the angular distance due to the semidiameter. This 
quantity, both for the sun and moon, is given in the Nautical Almanac. 
But in the case of the moon the semidiameter itself requires a small 
correction depending upon the observed altitude. For the semidiameter, 
furnished by the Nautical Almanac, is the apparent horizontal semi- 
diameter, or the angle it subtends when in the horizon; but as the moon 
approaches the zenith, her distance from the observer diminishes, and 
therefore her semidiameter is viewed under a greater angle. As she is 
nearer to the observer when in the zenith than when in the horizon, by 
one semidiameter of the earth, and as her distance from the earth's 
centre is about 60 semidiameters of the earth, the horizontal semidi- 
ameter will in the zenith become increased by about -g\th part, and at 
intermediate elevations the increase will be as the sine of the altitude. 
On this principle is formed the Table at the end, entitled Augmentation 
of the Moon's Semidiameter, and containing the proper correction to be 




NAUTICAL ASTRONOMY. 91 

added to the given horizontal semidiameter to obtain the true semi- 
diameter. 

On account of the great distance of the sun, no such correction of his 
semidiameter is necessary. 

The corrections for dip and semidiameter being thus applied, the re- 
sult is called the apparent altitude of the centre. In the case of the stars 
the only correction for the apparent altitude is the dip. It must, how- 
ever, be here remarked, that if the centre of the object were visible, and 
its altitude, instead of that of the limb, were to be taken, we should not, 
after applying the correction for dip, obtain precisely the same result 
as that which we have just called the apparent altitude of the centre, 
but should get a value somewhat less. The reason of this is, that every 
vertical arc in the heavens is shortened by refraction., as we shall shortly 
explain, so that the centre would not exceed the observed altitude of the 
lower limb, or fall short of that of the upper, by so great a quantity as 
the true semidiameter. Hence, from the apparent altitude of the centre, 
as found from applying the true semidiameter to the apparent altitude of 
the limb, a small quantity should in strictness be subtracted, and this 
small correction becomes necessary when the longitude is to be deter- 
mined with accuracy. This correction was first proposed by Dr. Thomas 
Young. A table for it is given at the end. 

To obtain the true altitude requires two other corrections, viz. for re- 
fraction and for parallax. The former of these has indeed an effect 
upon the two preceding corrections, dip, and semidiameter, which 
require certain modifications in consequence. One of these we have 
adverted to above, and the other will be noticed more particularly in 
the following article. 

Of Refraction, 

(77.) It is a universal fact in optics, that if a ray of light pass obliquely 
out of one medium into another of greater density, it will be bent out 
of its original direction at the point when it enters the new medium, 
and proceed through it in a direction more nearly perpendicular to its 
surface at that point. Hence the rays of light, proceeding from the 
celestial bodies, become bent downwards as soon as they enter the atmos- 
phere, their course being directed more nearly towards the centre of 
the earth, so that the rays which enter the eye of an observer, and by 
which any celestial object becomes visible to him, would, if not thus 
bent down, pass over his head ; the object is therefore seen by him above 
its true place : the angle between this apparent direction, and the true 
direction of the object, measures the refraction ; and, like the correction 
for dip, it is always subtractive ; it increases from the horizon, where 
it is greatest, to the zenith, where it vanishes, as the rays from objects 
in the zenith enter the atmosphere perpendicularly. 

It is the refraction which causes the sun and moon, when near the 
horizon, to present sometimes an elliptical appearance, the vertical dia- 
meter (and, indeed, every oblique diameter) seeming to be shorter than 
the horizontal, because the lower limb, or edge, being more elevated by 
refraction than the upper, the two are brought, in appearance, more 
nearly together. 

At the end of the volume we have given a table of refractions for 
different altitudes, from the horizon to the zenith, and adapted to the 
mean state of the atmosphere; but, as the actual state of the atmosphere 
generally differs from this, it becomes necessary, where the true alti- 
tude of the body is required with the utmost accuracy, to apply a cor- 
rection to the numbers in this table, so as to adapt them to the existing 
temperature and density of the atmosphere at the time of observation, 



92 PLANE AND SPHERICAL TRIGONOMETRY. 

as is indicated by the thermometer and barometer. The table of correo 
tions is annexed to the table of mean refractions. It should, however, 
be observed that below 4'^ the refraction is very variable and uncertain, 
and such low altitudes should be avoided as much as possible at sea. 

It will be unnecessary to use this annexed table for correcting the 
altitude of a celestial object when the latitude of the ship is the only 
object of the observation, as such a correction could seldom make a 
difference so great as half a mile in the resulting latitude ; but in de- 
termining the longitude by the Ijiinar observations, the neglect of these 
small corrections would sometimes introduce an error in the resulting 
longitude of more than 30 miles. 

It should be remarked here, that the dip, as determined in article (75), 
is on the supposition that refraction has not elevated the apparent hori- 
zon, but as such is not the case, the dip requires a correction; the 
amount of this correction is very uncertain, on account of the irregu- 
larity of the horizontal refractions although it is unquestionable that 
some correction is requisite. It is usual to allow about I or -^ of the 
computed dip for the correction. In our table -^ is allowed, which is 
according to Dr. Maskelyne, but Lambert and Legendre make it -fV- 

When the foregoing corrections have been applied to the observed 
altitude, the result will be the true altitude of the centre above the visi- 
ble horizon, and it remains now to apply the correction necessary to 
reduce this to the true altitude of the centre above the rational horizon ; 
that is, to the altitude which the body would have if the observer were 
situated at the centre of the earth instead of on its surface. 

Of the. Parallax. 

In order to explain the nature and effect of pa- 
rallax, let S represent the place of the object ob- 
served from the surface of the earth, at E ; then 
the angle SEH, that is, the observed angle when 
corrected for dip semidiameter, and refraction, 
will be the true altitude of the object, in reference 
to the observer's sensible horizon EH; and the 
angle SCR will be the true altitude, in reference 
to the rational horizon CR; and the difference of 
these angles is the parallax in altitude. If the 
body be at H, in the sensible horizon, then the difference of which we 
speak is the entire angle HCR; this is called the horizontal parallax. 

Since the angle SE'H is equal to the angle SCR, we have for the 
parallax in alt., SE'H — SEH = ESC ; that is, the parallax is the 
angle which the semidiameter of the earth subtends at the object ; it is 
obviously greatest in the horizon, and nothing in the zenith, and is the 
quantity which must be added to the true altitude above the sensible 
horizon, to obtain the true altitude above the rational horizon. 

The sun's parallax in altitude is given in a Table at the end ; and the 
moon's horizontal parallax is given for the noon and midnight at 
Greenwich) of every day of the year, in the Nautical Almanack: and 
from the horizontal parallax thus obtained the parallax in altitude must 
be calculated. This is easy; for since in the triangle SEC, Ave have 
the proportion SC : EC :: sin. SEC = sin. SEZ ^ cos. SEH : sin ESC ; 
it follows that the sine of the parallax in altitude varies as the cosine of 
the altitude, so that, as rad. is to the cosine of the altitude, so is the sine 
of the horizontal parallax, to the sine of the parallax in altitude. In 
other words, the log. sine of the horizontal parallax, added to the log, 
cosine of the altitude, abating 10 from the index, will give the log. sine 




NAUTICAL ASTRONOMY. 93 

of the parallax in altitude ; but as the parallax is ahvays a very small 

angle it is usual to substitute the arc for its sine, so that 

log. hor. par. in seconds -j- log. cos. alt. — 10 = log. par. in alt. in seconds. 

We must observe here that the horizontal parallax, given in th(; 
Nautical Almanack, is calculated to the equatorial radius of the earth ; 
and, therefore, except at the equator, a small subtractive correction will 
be necessary, on account of the spheroidal figure of the earth. A table 
of such corrections is given at the end, and explained in the introduc- 
tion to the tables. 

Such are the corrections necessary to be applied to the observed 
altitudes of celestial objects in order to obtain their true altitudes. A 
few other preliminary, but very simple, and obvious operations mast 
also be performed upon the several quantities taken out of the Nautical 
Almanack, in order to reduce them to their proper value at the time and 
place of observation ; for the elements furnished by the Nautical Alma- 
nack are computed for certain stated epochs, and their values for any 
intermediate epoch must be found by proportion. But ample directions 
for these preparatory operations are contained in the " Explanation of 
the Articles in the Nautical Almanack," by the late Dr. Maskelyne, 
which accompanies every edition of that work. 

Example of the Corrections. 

1. On the 14th of January, 1833, suppose the observed altitude of the 
sun's lower limb to be 16° 24', the observer's eye to be 18 feet above the 
level of the sea, the barometer to stand at 29 inches, and the thermo- 
meter at 58° : required the true altitude of the sun's centre. 

Observed alt. ©'s L. L. - 16° 36' 4'' 

Depression of the horizon - — 4 4 



App. alt. of L. L. - - - - 16 32 

Refraction - - - - - — 3 14 

Correction for Barometer - — 6*5 

Correction for Thermometer - - — 3-2 



True alt. of L. L. above visible horizon 16 28 36-3 
Sun's semidiameter (Naut. Aim.) -j- 16 17'3 

Parallax in altitude _ - - _|_ 8-4 



True altitude of Sun's centre - 16 45 2. 

2. On the 20th of May, 1833, suppose that in longitude about 77° 30' 
west, and lat. about 48° north, at 3'' apparent time, the altitude of the 
moon's lower limb is observed to be 18'' 8' 34", the height of the eye 
being 20 feet, the barometer 28-5 inches, and the thermometer 46° : re- 
quired the true altitude of the sun's centre. Here the object being the 
moon, it will be necessary to compute the parallax in altitude, from 
having the horizontal parallax corresponding to the time at Green v\^ich. 

The horizontal parallax is given in the Nautical Almanack for every 
noon and midnight ; and, therefore, to find it for any other intermediate 
time, we must say as 12'*^ is to its variation in 12'', so is the proposed 
time to the variation due to that time. 

In like manner must the moon's semidiameter be reduced, by propor- 
tion to the time of observation, since it sensibly varies in the course of 
a few hours. We shall begin, therefore, with finding in this way the 
true horizontal parallax and semidiameter for the time of the observa- 
tion reduced to the meridian of Greenwich. 



94 



PLANE AND SPHERICAL TRIGONOMETRY. 



Longitude of the ship in time o^ 10'" after Greenwich time. 
Apparent time at ship 3 

Apparent time at Greenwich 8 10 



Hor, par. at noon (Naut. Aim.) 58' 17" 
Hor. par. at midnight . 58 31 

Variation in 12'' 



.-.12^: 8''10'":: 14": 
Hor. par. at noon 

Hor. par at reduced time 



Ditto in seconds 

Dim. of par. for lat. 48° 

Tnie hor, parallax 



14 

9-5 

58' 17 



58 26-5 
60 



3506-5 
— 6-3 



3500-2 



Semidiam. at noon 15' 53" 

Semidiam. at midnight 15 57 

Variation in 12^^ 4 

.-. 12^;8''10'"::4": 2-7" 
Semidiameter at noon 15' 53 
Hor. semidia. at re 

duced time 15 55-7" 

Augmentation for 18=" 

alt. 5-2 



True semidiameter 16 0*9 



For the Apparent Altitude. 

Observed altitude of ]) s' L. L. . 18° 8' 34'' 

Depression , . . — 4 17 

Semidiameter minus contraction 15 57'9 



Apparent alt. ]) 's centre 18 20 14-9. 
For the Pa,ro2lax in Altitude. 
COS. ])'s app. alt. 18° 20' 15" . 99773668 
hor. parallax 35002" log. . 3-5440929 


Par. in altitude 3322-5" 

For the true Altitude. 
Apparent alt. of ]) 's centre 
Refraction 
Barometer 
Thermometer 


3-5214597 

18=^ 20' 14-9" 
— 2 54-2 
— 8-8 
-f 1-4 



True alt. above sensible horizon 18 17 13-3 

Parallax in altitude 3322-5" = 4-55 22-5 



True alt. of ]) 's centre 19 12 35-8. 

These two examples will serve for specimens of the corrections to be 
applied to an observed altitude, in order to deduce from it the true alti- 
tude of the body's centre. In the case of the moon, the corrections, 
when the utmost accuracy is sought, are rather numerous, as the last 
example shows. But in finding the latitude at sea, it is usual to dis- 
pense with some of these, more especially with the corrections for 
temperature, for the contraction of the moon's semidiameter, and for 
the spheroidal figure of the earth; because an error of a few seconds in 
the true altitude will introduce no error v/orth noticiag in the resulting 
latitude. When, however, the object of the observer is to deduce the 
longitude of the ship, all the data, furnished by observation, should be as 
accurate as possible ; for the problem is one' of such delicacy that by- 
neglecting to allow for the influence of temperature would alone intro- 
dace in some cases an error of from 30 to 40 miles in the longitude. 

When the object observed is a star, several of the foregoing correc- 
tions vanish : the only corrections, in this case requisite, are those for 
dip and refraction, modified as usual for the temperature. 



NAUTICAL ASTRONOMY. 95 

(78.) To determine the latitude at sea from the meridian altitude ofatvy 
celestial object lohose declination is known. 

The determination of the latitude, by a meridian altitude, is the 
easiest, and in general the safest, methoa of finding the ship's place on 
the meridian; for both the observations and the subsequent calculations 
being few, they are readily performed, and with but little liability to 
error in the result ; this method, therefore, is always to be used at sea, 
unless foggy or cloudy weather render it impracticable. 

The declination of the object observed is supposed to be given in the 
Nautical Almanack for the meridian of Greenwich ; it may therefore be 
reduced to the meridian of the ship by means of the longitude by ac- 
count, which will always be sufliciently accurate for this purpose, al- 
though it should differ very considerably from the true longitude, because 
declination changes so slowly that even an error of an hour in the longi- 
tude would cause an error in the declination too small to deserve notice. 

Having then thus found the distance of the object from the equinoc- 
tial, and having, by means of the observed altitude properly corrected, 
obtained the distance of the same object from the ship's zenith, the dis- 
tance of the zenith from the equinoctial, that is, the latitude, imme- 
diately becomes known. 

1. Let S be the object observed, the zenith Z being to the north of it, 
and the object itself north of the equinoctial EQ,, then the latitude EZ 
is equal to the zenith distance, or coaltitude ZS -[- the declination, and 
it is north. 

2. Let S' be the object, still north of the equinoctial, but so posited 
that the zenith is south of it, then the latitude EZ is equal to the dif- 
ference between the zenith distance S'Z, and declination S'E, and is 
still north. 

3. Let now the object be at S'^, south of the equir 
noctial, and the zenith to the north of the object, 
then the latitude EZ is equal to the difference be- ^^ 
tween the zenith distance S"Z and declination S'^E, ^ 
and it is north. 

"We have here assumed the north to be the ele- 
vated pole, but if the south be the elevated pole, then 
we must write south for north, and north for south. 
Hence the following rule for all cases. Call the 
zenith distance north or south, according as the zenith is north or 
south of the object. If the zenith distance and declination be of the 
same name, that is, both north or both south, their sum will be the lati- 
tude ; but, if of different names, their difference will be the latitude, of 
the same name as the greater. 

EXAMPLES, 

1. If on the 2d of May, 1833, the meridian altitude of the sun's lower 
limb be 47° 18^ height of the eye 20 feet, and longitude by account 32° 
E. : required the latitude, the sun being south at the time of observation. 
Observed alt. of ©'s L. L. . 47° 18' 0'' 

Dip, of the horizon . . — 4 17 




App. alt. of 0's L. L. . 46 13 43, 

The longitude in time is 2'' S'" east, so that time at Greenwich is 2^ 
8"* before the noon of the 2d of May ; hence, to find the corresponding 
declination, we have, by the Nautical Almanack, 24^: 2^^ 8'" ::18' V: V 
38''; so that, 1' 38", the variation in 2^* 8™, must be subtracted from 15° 
23' 21" N., the declination of the sun on May 2, at noon: hence the 
proper declination is 15° 21' 43" N. 



96 PLANE AND SPHERICAL TRIGONOMETRY. 

Observed alt. of 0's L. L. 
Dip. 

App. alt. of 0's L. L. 

Refraction 

Parallax 

Semidiam. (Naut. Aim.) 

True alt. of Q's centre 
Zenith distance - - - 
0's declination 

Latitude 58 52 57 N. 

2. On the first of January, 1820, the meridian altitude of Capella was 

27° 35', the zenith being south of the star, and the height of the eye 22 

feet ; required the latitude, 

Observed altitude - 27° 35' 0'' 

Dip - — 4 30 

Apparent altitude 
Refraction - - - 



47 


n8' 


0'' 


— 


4 17 


46 


13 43 







56 




+ 


6 




15 53 


46 


28 


46 


43 


31 


14 N. 


15 


21 


43 N. 



27 30 30 
— 1 51 


27 28 39 
62 31 21 S. 
45 48 39 N 



True altitude - - - 
Zenith distance - - - 
Star's dec. (Naut. Aim.) 

Latitude - - - 16 42 42 S. 

3. On the 19th of February, 1823, the ship being in longitude 40° W., 
the observed meridian altitude of the moon's lower limb was 55° 6' ; the 
zenith north of the moon ; and the height of the eye 16 feet : required 
the latitude. 

Here the time of observation at the ship is not given, it must there- 
fore be calculated, a,nd we have these data for this purpose, viz. that the 
ship is 40° W. of Greenwich, and that the moon is on its meridian. 
The following process therefore immediately suggests itself. 
The moon passed the merid. of Greenwich Feb. 19 (Naut. .Aim.) S'' 56'" 

Feb. 20 - 7'59 



Interval between the two passages - - 24 -j- 1 3 0. 

Hence P3'"is the moon's retardation in 25^^ 3™, and by proportion 
using for the longitude 40° W., its value in time 2'' 40'", we have, 

25/t ^m . ih^m .. 2f^ 40^" : O'' 6^ 42^ ; 
that is, the moon is retarded 6"^ 42® in passing from the meridian of 
Greenwich to that of the ship, and, therefore, instead of the apparent 
time at the ship being 6'' 56^", as it necessarily would be if there were no 
retardation, it will be 6^" 42^' later. Hence 

Apparent time at the ship - 7'' 2'" 42^ 

Ship's longitude W. - - 2 40 

Time at Greenwich - - 9 42 42. 

Having thus got the apparent time at Greenwich when the observa- 
tion was made, we may, by a reference to the Nautical Almanack and a 
subsequent proportion, find the moon's declination at that time : thus 
Moon's declination at Greenwich, Feb. 19 at noon 26° 38' 17" 

Feb. 19 at midn. 26 54 39 



Change of declination in 12 hours - - - 16 22, 

.-. 12'' : 9^ 42'" 42* :: 16' 22" : 13' 15" ; 



NAUTICAL ASTKONOBIY. ' 97 

hence 13' 15'' is the amount of the change of declination, from noon to 
g"* 43'", on the supposition, however, that the motion of the moon in de- 
clination may be considered as equable during the twelve hours. But 
on account of the irregular motion of the moon, this supposition intro- 
duces a sensible error, which may however be corrected by means of the 
table of" Equation of second Differences," given in the Nautical Alma- 
nack, and explained in Dr. Maskelyne's accompanying " Explanation.' 
The correct change of declination is thus found to be 14' 16". But from 
the year 1833, the declination of the moon will be given in the Nautical 
Almanack to every three hours, and the change for any shorter interval 
may then be obtained with the requisite accuracy by proportion, as 
above. Taking in the present case 14' 16" for the correct change, we 
have 

Declination for preceding noon 26° 38' 17" N. 

Increase of Declination - 14 16 



Declin. at the time of observation 26 52 33 N. 

Before we can find the proper correction for parallax, we must de- 
duce the apparent altitude of the centre, 

Observed altitude of ]) 's L. L. - 55° 6' 0" 

Dip. - - - - _ _ 3 50 

Semidiameter (Naut. Aim.) - - 16 13 

Augmentation for 55° of alt. - - 13 

Apparent alt. of D's centre - - 55 18 36 cos. 9-7552161 

Hor. par. in seconds at 9^ 43^^ (Naut; Aim.) 3572 log. 3-5.529115 

Parallax in altitude in seconds - 2033 log. 3-3081276 

therefore the correction for parallax is 33' 53". 
Having thus reduced all the corrections to the time of observation, 

we readily obtain the true altitude, and thence the latitude as follows, 
Apparent alt. of J) 's centre 55° 48' 36" 

Refraction - - - — 40 

Parallax in altitude . _ _ 33 53 



True altitude - - 55 51 49 

Zenith distance - - 34 8 UN.) 

Declination - - - 26 52 33 N. 5 



Latitude - - - 61 44 N. 



These examples will, no doubt, be found sufficient to put the student 
in possession of the method of applying the various corrections to the 
observed meridian altitude of a celestial object, in order to deduce from 
it the latitude of the ship. But it should be remarked, that in most 
works on Nautical Astronomy, subsidiary tables are inserted for the 
purpose of abridging some of the foregoing corrective operations ; such 
tables, therefore, offer very acceptable aid to the practical navigator. 
The most esteemed works of this kind are Dr. Mackay's '* Treatise on 
the Theory and Practice of finding the Longitude at Sea ;" the "Nau- 
tical Tables" of J. De Mendoza Rios, and Mr. Riddle's book on Navi- 
gation and Nautical Astronomy. 

It should also be observed here, that in the preceding examples the 

celestial object is supposed to be on the meridian above the pole; that 

is, to be higher than the elevated pole. But, if a meridian altitude be 

taken below the pole, which may be done if the object is ciratmpolar, 

9 M 



98 PLANE AND SPHERICAL TRIGONOMETRY. 

or SO near to the elevated pole as to perform its apparent daily revolution 
about it without passing below the horizon, then the latitude of the place 
will be equal to the sum of the true altitude, and the codeclination or 
polar distance of the object ; for this sum will obviously measure the 
elevation of the pole above the horizon, which is equal to the latitude. 

(79.) To determine the latitude at sea, by means of two altitudes of the 
sun, and the time between the observations. 

In the preceding article we have shown how to determine the latitude 
of the ship by the meridian altitude of the sun, or of any other heavenly 
body, whose declination may be found. But, as already remarked, the 
object we wish to observe may be obscured when it comes to the meri- 
dian, and this may happen for many davs together, although it may be 
frequently visible at other times of the day. As therefore the opportu- 
nity for a meridian observation cannot be depended upon, it becomes an 
important problem to determine the latitude at sea, by observations 
made out of the meridian ; and considerable attention has accordingly 
been paid, by scientific persons, to the method of finding the latitude by 
" double altitudes," and various tables have been computed to facilitate 
the operation. But the direct method, by spherical trigonometry, 
though rather long, involving three spherical triangles, will be more 
readily remembered, and more easily applied by persons familiar with 
the rules and formulas of Trigonometry, than any indirect or approxi- 
mative process ; we shall therefore explain the direct method. 

Let P be the elevated pole, Z the zenith of the 
ship, and S, S' the two places of the sun when the alti- 
tudes are taken. Then, drawing the great circle 
arcs as in the figure, we shall have these given quan- 
tities, viz. the codecliiiations PS, PS'; the coaltitudes| 
ZS, ZS', and the hour angle SPS', which measures 
the interval between the observations ; and tlie quantity 
sought is the coaltitude ZP. Now, in the triangle 
PSS', we have given two sides and the included angle 
to find the third side SS', and one of the remaining angles, say the 
angle PSS'. In the triangle ZSS' we have given the three sides to 
find the angle S'SZ ; having then the angles PSS', S'SZ, the angle 
ZSP becomes known, so that we have lastly, two sides and the included 
angle in the triangle ZSP, to find the third side ZP. 

Before the application of the trigonometrical process, the observed 
altitudes must, of course, be reduced to the true altitudes, as in the pre- 
ceding examples. Moreover, as the ship most probably sails during 
the interval of the observation, an additional reduction becomes neces- 
sary, the first altitude must be reduced to what it would have been if 
taken at the place where the second was taken : this correction will be 
known if we know the number of minutes the ship has sailed directly 
towards or directly from the sun, upon leaving the place where the first 
observation was made. To find this, take the angle included between 
the ship's course and the sun's bearing, at the first observation ; and con- 
sidering this angle as a course, and the distance sailed as the corres- 
ponding distance, find by the traverse table, or by the operation of plane 
sailing, the difference of latitude, which will be the amount of the 
approach to, or departure from, the sun. This must be added to the 
first altitude if the angle is less than 90°, because the ship will have 
approached towards the sun ; but it must be subtracted when the 
angle exceeds 90°. If the angle is 90° no correction for the ship's 
change of place will be necessary. 

The truth of this correction will be immediately seen by considering 
that if the sun's centre were the elevated pole, what is in reality the 




NAUTICAL ASTRONOMY. 99 

coaltitude would then be the colatitude, and, therefore, that, by what- 
ever quantity this latter is increased or diminished by the ship's motion, 
on the one hypothesis, by the same quantity will the former be increased 
or diminished on the other hypothesis. 

Where great accuracy is aimed at, account should be taken of the 
ship's change of longitude during the interval of the observations ; 
when converted into time it must be added to the interval of time 
between the observations when the ship has sailed eastward, and sub- 
tracted when she has sailed westward. This correction is very easily 
applied. 

Having thus mentioned the necessary preparative corrections, we 
shall now give an example of the trigonometrical operation. 

EXAMPLES. 

Let the two zenith distances corrected be (see last fig.) ZS = 73^ 54' 
13'', ZS' = 47° 42' 51", the corresponding declinations 8^ 18' and 8° 15' 
north, and the interval of time three hours ; to determine the latitude. 

Considering SS' to be the base of an isoceles spherical triangle, of 
which one of the equal sides is i{FS -f- PS') = 81° 43' 30", and the 
vertical angle equal to 3^^ or 45^, let the perpendicular PM be drawn, 
then we have in the triangle PMS right-angled at M, PS = 81° 43' 30" 

45° 
and P = —r- = 22° 30' j given to find SM = | SS' as follows. 

1. To find SS'from the triangle PMS. 
sin. PS 81° 43' 30" . . . 9-9954547 

sin. P 22 30 . . . 9-5828397 



sin. SM22 15 11-3 . . 95782944 

2 



SS' = 44 30 22-6. 

n. To find PSS' from the triangle PSS'. 
sin. SS' 44° 30' 22-6" - arith. comp. 0-1542898 

sin. PS' 81 45 - - - 9-9954822 

sin, SPS'45 - - _ 9-8494850 



sin. PSS' 86 38 58 - - - 9-9992570. 

This angle is acute like its opposite side, (see art. 60.) 

m. To find ZSS' m the triangle ZSS' 
ZS' 47° 45' 51" 
sm. ZS 73 54 13 - arith. comp. 00173686 

sin. SS' 44 30 22-6 - arith. comp. 0-1542898 

2)166 10 26-6 

J Sum = 83 5 13-3 
sin. (§ Sum— ZS) 9 11 0-3 - - - 9-2030206 

sin. (i Sum — SS') 38 34 50-7 - - - 9.7949179 

2)19-1695969 

sin. I ZSS' 22 36 26-4 - - 9-5847985 

.-. ZSS' = 45° 12' 52-8" 

PSS' = 86 38 58 



PSZ == 41 26 5-2. 



100 PLANE AND SPHERICAL TRIGONOMETRY. 

IV. To find ZF in the triangle "L^V . 

10-8359917, COS. PS - 9-1594354 

9-8748930, sin. w, ar. comp. -7189551 

10-7108847, sin. (w-f-ZS) 9-9982874 

sin. 48^ 49' 59-7^' 9-8766779 



tan. PS 81° 
COS. PSZ 41 


4-2' 

26 


0" 
5-2 


cot. w 11 





41-2 


73 


54 


54-2 



o -f ZS = 84 54 54-2. 

Hence the latitude is 48° 50'. 

2. Tne two corrected altitudes are 42° 14' and 16° 5' 47", the cor- 
responding declinations 8° 16' 30" and' 8° 15', and the time between 
the observations 3 hours ; required the latitude of the place. 

The latitude is 48° 54' 27" N. 

Upon the same principle may the latitude be determined from the 
altitudes of two fixed stars, taken at the same time ; in this case S, S', 
in th? preceding figure, will represent the two stars; PS, PS', their 
known polar distances, and the angles SPS'the difference of their right 
ascensions ; the same quantities are therefore given as in the case of the 
sun, but as in the case of two stars PS, PS', may differ Very consider- 
ably ; SS' cannot be considered as the base of an isosceles triangle, but 
must be computed from the other two sides and their included angle. 
In the Nautical Almanack for 1825 Dr. Brinklej'- has computed for 
1822, and tabulated, the distances SS' for certain pairs of stars, conve- 
niently situated for observation, and has annexed the change of dis- 
tance corresponding to 10 years. The same table shows also the dif- 
ference of right ascension for each pair of stars, with the change in 10 
years ; so that by help of this table the computation for finding the lati- 
tude from the simultaneous altitudes of two fixed stars becomes consi- 
derably abridged. 

For other methods of determining the latitude, the student may con- 
sult " Mackay on the Longitude," vol. i., and Captain Kater's Nautical 
Astronomy, in the Ency. Metropolitan, &c. 

On finding the Longitude by the lAinar Observations. 

(80.) There are several astronomical methods of determining the lon- 
gitude of a place, which cannot be accurately employed at sea, on account 
of the great difficulty of managing a telescope on shipboard ; we shall 
not, therefore, enter here into any explanation of these methods, but 
shall confine ourselves to the Innar method of determining the longitude, 
which is justly regarded as the principal problem in Nautical Astro- 
nomy. Before entering upon the solution of this problem it will be 
necessary to make a few introductory remarks. 

The determination of the longitude of a place always requires the 
solution of these two problems, viz. 1st, to determine the time at the place 
at any instant ; and, 2d, to determine the time at the first meridian, at 
the same instant ; for the difference of the times converted into degrees, 
at the rate of 15° to an hour, will obviously give the longitude. 

When the latitude of the place is known, (and it may be foimd by the 
methods already explained,) the time may be computed from the alti- 
tude of any celestial object whose declination is known ; for the coalti- 
tude, codeclination, and colatitude, will be three sides of a spherical 
triangle given to find the hour angle, comprised between the codeclina- 
tion and the colatitude. But to find the time at Greenwich requires the 
aid of additional data, besides those furnished by observations made at 
the place. The Greenwich time may, indeed, be obtained at once, inde- 
pendently of any observations at the place, by means of a chronometer, 



NAUTICAL ASTRONOMY. 101 

carefully regulated to Greenwich, time, provided it be subject to no irre- 
gularities after having been once properly adjusted. A ship furnished 
with such a timepiece always carries the Greenwich time with her,* 
and the longitude then becomes reduced to the problem of finding the 
time at the place. Chronometers are now brought to such a state of 
perfection that very great dependence can be placed on them, and they 
are accordingly always taken out on long voyages for the purpose of 
showing the Greenwich time, and are thus of great use to the mariner. 
Still, however, as the most perfect contrivance of human art is subject 
to accident, and the more delicate the machine the more liable is it to 
disarrangement, from causes which we may not be able to control, it 
becomes highly desirable, in so important a matter as finding the place 
of a ship at sea, to be possessed of methods altogether beyond the influ- 
ence of terrestrial vicissitudes, and such methods the celestial motions 
alone can supply. The angular motion of the moon in her orbit is 
more rapid than that of any other celestial body, and sufficiently great 
to render the portion of its path passed over in so short a time as two 
or three seconds, a measurable quantity even with a small portable 
instrument (the sextant). 

It is obvious, therefore, that if the distance of the moon's centre from 
any celestial body, in or near her path, be computed for any Greenwich 
time, and this distance be found the same as that given by actual obser- 
vation at any place, then the difference between the time of observing 
the phenomenon and the time at Greenwich, when it was predicted to 
happen, will give the longitude of the place of observation. Now in 
the Nautical Almanack the distances of the moonfrom the sun, and frorr 
several of the fixed stars near her path, are given for every three hourt: 
of apparent Greenwich time, and for several years to come ; and the 
Greenwich time, corresponding to any intermediate distance, is obtained 
by simple proportion, with all requisite accuracy ; so that by means of the 
Nautical Almanack we may always determine the time at Greenwich 
when any distance observed at sea was taken. 

The distances inserted in the Nautical Almanack are the true angular 
distances between the centres of the bodies, the observer being consi- 
dered as at the centre of the earth, and to the true distance therefore 
every observed distance must be reduced ; it is this reduction which 
constitutes the trigonometrical difficulties of the problem ; and it con- 
sists in clearing the lunar distance from the effects of parallax and refrac- 
tion ; how to do this it is now our business to explain. 

Let m, 5, be the observed places of the moon and Z 

sun, or of the moon and a fixed star, and let M, S, "^ 

be their true places. M will be above m, because 
the moon is depressed by parallax more than it is 
elevated by refraction ; but S will be below 5, be- 
cause the sun is more elevated by refraction than 
it is depressed by parallax. Observation gives the 
apparent distance ms^ and the apparent zenith 
distances Zm, Zs: by applying the proper correc- 
tions to these latter we also deduce the true zenith 
distances ZM, ZS, and with these data we are to determine the true 
distance, MS, by computation. 

Put d for the apparent distance, 
D true distance, 

a, a' apparent altitudes. 

A, Af true altitudes. 

* As chronometers show mean time, the equation of time must be applied to obtain the 
apparent time at Greenwich. 
9* 




102 PLANE AND SPHERICAL TRIGONOMETRY. 

COS. D — sin. A sin. A' 



Then in the triangle MZS, we have cos. Z = - 



COS. A COS. A' 
, . , . ■, rr n- cos. d — sin. a sin. a' 

and in the triansie mZs, cos. Z = ; : 

COS. a cos. a^ 
hence, for the determination of D, we have this equation, viz. 
cos. D — sin. A sin. A' cos. d — sin. a sin. a' 



cos. A cos. A' COS. a cos.. a' ' 

from which we immediately get 

T^ / 7 • • ,N COS. A COS. A^ , . . . . , 

COS. D = (cos. d — sm. a sm. « ) h sm. A sm. A 

COS. a cos. a' 
cos.d-\-cos.(a-{~a^) — cos. a cos. a' . ,,.».*, 

— ■ ^ — ^ COS. A cos. A' + sm. A sm. A' 

COS. a COS. o/ 

2cos.^{a-\-a'-{-d)cos.i(a-4-a'-^d)cos.Acos.A' ,^ , .„,.. 

= 1 ! — ^- ^-^ — ! '- cos.(A+ AO.(l) 

COS. a COS. a' 

.2cos. ^(a-f-^'H- ^)cos.H«+«''*'^)cos. Acos.A' > /a laa 

= J ; 7-r ; --— 1 J COS.( A+A )J 

< COS. ft COS. «' COS. (A -|- A') ' 

or calling the first term within the brackets 2 cos.^ F, 
COS. D = (2 cos.2 F — 1) cos. (A 4- A') = cos. 2 F cos. (A -f A') . (2). 

The formulas marked (1) and (2) are both of them convenient for the 
computation of D ; a third formula may be obtained from (1), as follows. 
Subtract each side of (1) from 1 ; then since (p. 37,) 

1 — COS. D = 2 sin.2 10,1 + cos. (A + A') = 2 cos.2 |(A + A'), 
we have, after dividing by 2, 

. „,„ -,,. , .. , cos.i(«+a'+^)cos.|(«+^''*'<^)cos. Acos.A' 

sm.2 J D = 00S.2 1 ( A+ AO -— ■ — — ■ • 

COS. a COS. a' 

— Po«2lrAa-A'^J^ cos.^(g-f a^+tZ) cos. Ka+^'-^^) cos. Acqs. A^ ^, 

-COS. .CA+A)^1 ^s.acos.^'cos.fKA + AO ~^* 

or, calling the second term within the brackets sin.^ d, 
sin.2 i D = C0S.2 HAH- AO cos.2 6 
.-. sin. i D = COS. KA + A') cos. . . . . (3). 
This latter is Borda's formula. 
We shall solve an example by each of these formulas. 

EXAMPLES. 

1. Suppose the apparent distance between the centres of the sun and 
moon to be 83° 57' 33'', the apparent altitude of the moon's centre 27° 
34' 5", the apparent altitude of the sun's centre 48° 27' 32", the true 
altitude of the moon's centre 28° 20' 48", and the true altitude of the 
sun's centre 48° 26' 49" ; then we have d = 83° 57' 33", a = ^1° W 5" 

a' = 48° 27' 32" ; A = 28° 20' 48", A' = 48° 26' 49" ; 
and the computation for D, by the first formula is as follows : 
( d 83055' 33" 

comp. COS. 0523390 

comp. COS. 1783835 

log. 2 3010300 

COS. 9-2399686 

COS. 9-9989587 

- COS. 9-9445275 

COS. 9-8217187 





U 


27 34 


5 




\a' 


48 27 


32 




2)159 59 


10 


I sum 




79 59 


35 


1 sum -V. d 


3 57 


58 




A 


28 20 


48 




A' 


48 26 


49 



(Reject 40 from index) 1-5369260 = log. -3442921 + 
A 4- A' 76 47 37 - - - nat. cos. -2284595 ~ 



True distance 83° 20' 54" nat. cos. -1158326. 



NAUTICAL ASTKONOMY. 103 

By glancing at the formula (1), we see that 30 must be rejected from 
the sum of the above column of logarithms, so that the logarithmic line 
resulting from the process is 9-5369260. Now, as in the table of log. 
sines, log. cosines, &c., the radius is supposed to be 10'*^, of which the 
log. is 10, and in the table of natural sines, cosines, &c., the rad. is 1, 
of which the log. is ; it follows that when we wish to find, by help of 
a table of the logarithms of numbers, the natural trigonometrical line 
corresponding to any logarithmic one, we must diminish this latter by 
10, and enter the table with the remainder. Hence the sum of the fore- 
going columns of logarithms must be diminished by 40, and the re- 
mainder will be truly the logarithm of the natural number represented 
by the first term in the second member of the equation (1). If this 
natural number be less than nat. cos. (A -f A'), which is to be sub- 
tracted from it, the remainder will be negative, in which case D will 
be obtuse. 

By the second formula the process is as follows : 
d 83° 57' 33'' 

« 27 34 5 - - comp. cos. 00523390 

a' 48 27 32 - comp. cos. 01 783835 

2)159 59 10 



I sum 79 59 35 cos. 92399686 

i sum -^ <^ 3 57 58 - - cos. 9-9989587 

A 28 20 48 - ' - cos. 9-9445275 

A' 48 26 49 - - cos. 9-8217187 

A + A' 76 47 37 - comp. cos. 0641 1909 



2)19-8770869 



F 29 46 3 - - cos. 9-9385434 



2 F 59 32 6 - cos. 9-705018^ + 



True distance 83° 20' 54" cos. 9-0638273. 
In adding up the logarithms to find cos. ^F, 20 must be rejected from 
the index ; and the logarithm marked — , is to be subtracted from 
that marked -{- . Moreover, if A -|- A' and 2 F are both acute or both 
obtuse, D will be acute, otherwise it will be obtuse. 
We shall now exhibit the process by Bordah formula, 
d 83° 57' 33" 

« 27 34 5 comp. cos. 00523390 

«' 48 27 32 comp. COS. 0-1783835 



cos. 9-2399686 
COS. 9-9989587 
COS. 9-9445275 
COS. 9-8217187 





sum 


2^ 


)159 


59 10 


\ 


79 


59 35 


f 


sum 


^ d 


3 


57 58 






A 


28 


20 48 






A' 


48 


26 49 



A+A' 76 47 37 - 2)39-2358960 

19-6179480 
HA -f A') 38 23 48| - - cos. 9-8941654 -f- 



31 57 531 - > sin. 9-7237826 



104 PLANE AND SPHERICAL TRIGONOMETRY. 

9 - - COS. 9-9285870 -{- 

J D 41° 40' 21" sin. 9-8-227524 

.-. D = 83 30 54, the true distance. 

An estimate may now be formed of the relative advantages of these 
three methods, as regards practical facilit}". We are inclined to pre- 
fer the iirst method, which vre believe is new, as fewer references to the 
tables are requisite, and as, moreover, there are no arithmetical operations 
required, besides those which are actually exhibited. The second and 
third methods seem to offer nearly equal advantages : in the first of 
these, however, it may be observed that the trigonometrical lines in- 
volved are all of one name, viz. cosines, and that the final reference to 
the tables gives the true distance instead of its half, as in the last 
m.ethod. 

Each of the foregoing processes may be shortened by using a 
subsidiary table, containing the various values of the expression 
cos. A COS. A' _, , ^, 

. Such a table computed to every degree of the moon's 

COS. a COS. a 

apparent altitude, and to every 10 seconds of her horizontal parallax, 
forms Table ix. of the Requisite Tables, published b}' order of the 
Commissioners of Longitude. But a more complete table of this kind 
is given in the second volume of Dr. Mackay's work, on the Longitude, 
If each number in tliis table were increased by the constant number 
■3010300, the table itself would become somewhat simplified, and the 
process of clearing the distance by our first method would be rendered 
remarkably short and convenient. 

The preceding example is taken from Woodhouse's Astronomy, part 
n., p. 859, where the day of observation is stated to be June 5, 1793. 
Now by the Nautical Almanack, for that year, we have 
Distance at 15^ 83^ 6' 1", Also at time of observation D = 83^ 20' 55" 
at 18^- 84 28 26 at 15^' D = 83 6 1 



Increase in 3'^ I 22 25. Incs. between 15" and time of obs. 14 54 
.-. 1^ 22' 25" : 14' 54" :: 3^- : 32^ 33^ 
Hence, when the observation was made, the apparent time at Green- 
vrich. was 15'* 32'' 33 . 

To find the time at the ship, requires that we know the latitude of 
the place and the srm's declination. The former, therefore, mast have 
been previously ascertained, and the latter may now be found by means 
of the apparent Greenwich time just deduced, and the Nautical Alma- 
nack. We shall suppose the latitude to be 10^ 16' 40" S. ; the stm's 
declination will be 23= 22' 23", and taking the true altitude of the sun 
= 48=' 46' 49", we shall thus have, in order to find the time, three 
sides of a spherical triangle to find an angle. The computation is as 
follows. 

coalt. 41^ 33' 11" 
sin. colat. 79 43 20 arith. comp. 0-0070251 
sin. sun's polar dist. 113 22 48 arith. comp. 0-0372078 

2)234 39 19 



117 19 39-5 
sin. 37 36 195 - - 97854864 
sin. 3 56 51-5 - - 8-8378712 



2)18-6675905 



NAUTICAL ASTRONOMY. 105 

sin. 12° 27^71'' - 9-3337902 



Hour angle = 24 54 35 = 1* 39"^ 38'3s in time. 



Time at Greenwich 15 32 33 



L. in time, reckoning westward. 13 52 54' 7. 

Or, subtracting this from 24 hours, we have 10^^ 7»« 4-3s, for the longi- 
tude easb^ in time, and therefore the longitude in degrees is 151° 46 
^\" E. 

2. Given the apparent altitude of the moon's centre 8° 26' 13'", the 
true altitude 9° 20' 45'', the apparent altitude of a star 35° 40', the true 
altitude 35° 38' 49", and their apparent distance 31° 13' 26" ; to deter- 
mine the true distance. The true distance is 30° 23' 56". 

Those who are desirous of entering more at large into the problem of 
the Longitude, and of becoming acquainted with the best methods of 
shortening the computation by the aid of subsidiary tables, may advan- 
tageously consult, besides the works already referred to, the GLuarto 
Tables of J. De Mendoza Rioz, Lynn's Navigation Tables, Captain 
Kater's Treatise on Nautical Astronomy, in the Encyclopaedia Metro- 
politana, Kerrigan's Navigator's Guide and Nautical Tables, and Dr. 
Myers's translation of Rossel on the Longitude. 

Variation of the Compass. 

(81.) We shall conclude this part of our subject by briefly considering 
the methods of finding the variation of the compass, or the quantity by 
which the north point, as shown by the compass, varies easterly or 
westerly from the true north point of the horizon 

The solution of this problem merely requires that we find by compu- 
tation, or by some means independent of the compass, the hearing of a 
celestial object, that we observe the bearing by the compass, and then 
take the dilFerence of the two. The problem resolves itself, therefore, 
into two cases, the object whose bearing is sought being either in the 
horizon or above it ; in the one case we have to compute its amplitude, 
and in the other its azimuth. 

The computation of the amplitude is simply determining the hypote- 
nuse of a right-angled triangle, of which one side is given, viz. the d-e- 
cltnation of the object, as also the angle opposite to it, viz. the colatitude. 
The computation of the azimuth requires the solution of an oblique 
spherical triangle, the three sides being given to find an angle ; the three 
given sides are the colatitude ; the zenith distance of the object and its 
polar distance : and the azimuth being measured by the angle at the 
■ zenith opposite the polar distance, this is the angle sought. We shall 
give an example in each of these cases of the problem. 



1. In January 1830, at latitude 27° 36' N., the rising amplitude of 
Aldebaran was, by the compass* E. 23° 30' N. , required the variation. 

By the Nautical Almanack the declination of Aldebaran is 16° 9' 37" 
N., therefore since Rad. X sin. dec. = sin. amp. X cos. lat, the com- 
putation is as follows. 



* The compass amplitude must be taken when the apparent altituae cf the object is 
equal to the depression of the horizon. 

N 



106 PLANE AND SPHEEICAL TRIGONOI^IETEY. 

sin. declination 16^ 9' 37'' - - 9Uioo^ 

COS. latitude 27 36 - - 9-9475335 

srn. Amplitude E. 18 18 17 N. - - 9-4970292 



Magnetic Amptitude E. 23 30 N. 

Variation 5 11 43. 
As the object is farther from the magnetic east than from the true 
east, the magnetic east has therefore advanced towards the south, and 
therefore the magnetic north towards the east ; hence the variation is 
5° ir43"E. 

2. In latitude 4S^ 50' north, the true altitude of the sun's centre was 
22^ 2', the declination at the time was 10^ 12' S., and its magnetic 
bearing 161=' 32' east. Required the variation. 
O's polar distance 100^ 12' 

sin. zenith distance 67 58 arith. comp. 0'0329363 
sia. colatitude 41 10 arith. comp. 0-1816080 



2)209 20 
sin. IS 104 40 - 99856129 

(iS — pel. dist.) 4 28 - 88914209 

2)19-0915781 
COS. 69^ 25' 40" - 9 5457895 



2 



O's true azimuth N. 138 51 20 E. 
Observed azimuth N. 161 32 E. 



22 40 40 West. 
The variation is west, because the stm's observed distance from the 
north, measured easterly, being greater than its true distance, intimates 
that the north point of the compass has approached towards the west. 

3. In latinide 48^ 20' north, the star Rigel was observed to set 9^ 50^ 
to the northward of the west point of the compass ; required the varia- 
tion, the declination of Rigel being 8=' 25' S. Variation 22^ 33' West. 

4. In latitude 50== 12' north, when the sun's declination was 11^ 28' 
53" N., its true altitude was fotmd to be 37^ 0' 16", and the observed 
azimuth S. 31° E. ; reqtured the variation of the compass. 

Variation 28° 2' West. 



PART TV. 



MISCELLANEOUS TRIGONOMETRICAL INQUIRIES. 

(82.) We now come to the final part of our subject, in which we propose 
to bring together several miscellaneous particulars which properly come 
under consideration in a treatise on Trigonometry. One or two of 
these, especially those which relate to certain compendious solutions of 
plane triangles, and to the trigonometrical lines of small arcs, might 
have been introduced much earlier, although we have preferred to post- 
pone their consideration for a supplementary chapter, agreeing with 
Woodhouse, that it is better for the student first " to attend solely to the 
general solutions, and to postpone to a time of leisure and of acquired 
knowledge the consideration of the methods that are either more ex- 
peditious or are adapted to particular exigencies. 



CHAPTER L 

ON THE SOLUTIONS OF CERTAIN CASES OF PLANE TRIANGLES, AND ON DETER- 
MINING THE TRIGONOMETRICAL LINES OF SMALL ARCS. 

PROBLEM I. 

(83.) Given two sides and the included angle of a plane triangle, to 
determine the third side, without finding the remaining angles. 

The general expression for the side c, in terms of the two sides fl, 6, 
and the included angle C, is (17), 

c2 = «2_|_j2_2«^,cos. G = {a — hf-\-^ab(X — QOs. C) 

^{a — hf -^^ab-^sm.^ hO = ia — bf \\-\ — -r-^-^^ sm?\0\. 

\a b) 

Assume the second term within the brackets equal to tan.^O then, 
since 1 + tan.^fi = sec.^e = '—- > we have c = (a — b^ '-~ • 

' COS.20 ^ COS. d 

Hence c is determined by these two formulas, viz. 
log. tan. e = log. 2-\-i log. « -|- « ^^S- ^ + log. sin. J C — log. {a — b) 
log. c — log. {a — b)-\-10 — log. cos. d. 

EXAMPLE. 

Given a = 562, b = 320, and C = 128° 4', to find c. 

log. 2 0-3010300 

i log. 562 1-3748681 

J log. 320 1-2525750 

log. sin. 64° 2' 9-9537833 

ar. comp. log. 242 7-6161846, log. 242 + 10 . .13-3838154 

log. tan. 6 10-4984410 .'. log. cos. 6 - 9-4807177 



log.c 80001 . . 2-9030977. 



108 



PLANE TEIAXGLES. 



PROBLEM 11. 



Given the logarithms of two sides of a plane triangle, as also the 
included angle, to determine the remaining angles. 
Let log. a, log. b, and C, be given. Suppose a greater than b, and 

assume r -— = tan. Q ; then tan. being greater than 1, e will exceed 

45°. Also (19.) tan. KA — B) 

°' — ^.^r^ * . , /-. tan. 0—1 
= — TT cot. J C = cot. i C = -— — - cot. \ C 

- h'^ 
= tan. (0 — 45°) cot. J C (p. 33). 

Hence, introducing the radius, A — B is determmed by these two 
formulas, viz. log. tan. = 10 +log. a — log. h 

log. tan. HA — B) = log. tan. (0 — 45-) ^- log. cot. i C — 10. 
Thus, taking the example in the last problem, we have 
10 -}- log. 562 . . 12- 7497363 

log. 320 . . 2-5051500 

log. tan. , 10-2445863 .-. = 60= 20' 35" 



.-. 0—450 = 150 20' 35^^ 
Again, log. tan. 15° SC 35" 9-4383476 
log. cot. 64 2 9-6875402 

log. tan. KA — B) 9' 1258878 .-. HA — B) = 7^ 36' 40' 

^(AH-B) = 25 58 

A r= 33 34 40 
B = 18 21 20. 

This method of determining the angles A and B will always be the 
shortest, when instead of their sides their logarithms are given. Thus 
the solution of problem x.,p. 31, becomes much facilitated by the appli- 
cation of this process, 

PROBLEM m. 

To determine the area of a plane triangle when any three parts 
except the three angles are given. 

1. Let two sides b^ c, and the included angle A, be given. (See fig. 
p. 17.) 

The area of the triangle is expressed by \ AB • CD ; but CD = AC 
sin. A ; hence the expression for the area, in terms of the given quan- 
tities, is Area — \bc sin. A. 

2. Let two angles, A, B, and the interjacent side c, be given. 
Then, since sin. C : sm. B : : c : &, 

sin. B 7 . « sin. A sin. B „ 

we have b = —. — - c .-. be sm. A = -. — c^ : 

sm. C sm. C 

, , . ^ , . . sin. A sin. B „ 

hence the expression for the area is Area = — — -: — - — c^. 

2 sm. C 

3. Let the three sides be given. 



By an, (20), sin. i A = j Si?^^IM:Zi) , COS. iA = J tl«lr^ 

DC DC 



•. 2sin.|Acos, i A, or (art, 31) sin, A = T-ViS (IS — a) (iS— ^')(JS— c).- 
10 



NAUTICAL ASTRONOMY. 109 

Consequently, by substituting this value of sin, A in the first expres- 
sion, we have. Area =. \4 S (iS — a) (|S — *) (^ S— c); 
which formula furnishes the well known rule, given in all books on 
mensuration, for the area of a triangle when the three sides are given. 
(See Geom. p. 202,) These expressions for the area of a plane triangle 
are all adapted to logarithmic computation. 

PROBLEM IV. 

To find the logarithmic sine of a very small arc. 

By article (30) the expression for the sine of any arc x is, 

sin. x = x ' -|- — ' &c. Now as the length of 

an arc of one degree is -01745329, (see p. 36-7,) it is plain that, even when 
X is so great as this, the third term of the above series can have no sig- 
nificant figure in the first ten places of decimals. 
Retaining therefore only the first two terms, we have, when x is small, 

sin, :. = ^-p|^ = :.(l-2^'3-) = :.|l-iJ+2^S* nearly; 

that is, (p. 36,) sin. x = x cos." x\ hence, by introducing the radius, 
log. sin. X = log, X — \ (10 — log. cos. .^) . . . . (1). 

Let the arc x contam n seconds, then x = v 60 v fiO * 
hence, by introducing the radius, 



log, X = log, n + log, 3-14159, &c, + 10 — log. 180 X 60^ 
= log, n + 4-6855749; therefore, from (1), 
log, sin, X = log. n -{- 4-6855749 — | arith, comp, log, cos, x . . . (2) ; 
hence this rule. To the logarithm of the arc reduced into seconds, 
with the decimal annexed, add the constant quantity 4-6855749, and 
from the sum subtract one third of the arithmetical complement of the 
log, cosine ; the remainder will be the logarithmic sine of the given arc. 
This rule will determine the log. sine of a very small arc with great 
accuracy ; it was first given, without demonstration, by Dr. Maskelyne, 
in his Introduction to Taylor's Logarithms. The above proof is from 
Woodhouse's Trigonometry, 

PROBLEM V, 

To find the logarithmic tangent of a very small arc. 

Let X be the arc ; then, as we have found in last problem, 

1 sin. X .'c ^T. . , . , 

sm. = X cos.^ X .-. = tan. x = ~— ■ Hence, mtroducmg the 

cos. X gQg_ s ^ 

radius, log. tan. x •— log, .^ -f- I (10 — log. cos, x). 

The second member of this equation is equal to the second member 
of (1) in last problem, pins the arithmetical complement of log. cos. x ; 
hence, since the second member of (2) is equivalent to t!ie second mem- 
ber of (1), we have 

log. tan. X = log. n -\- 4-6855749 -f- § arith. comp. log. cos. x ... (3) ; 
which furnishes this rule. To the logarithm of the arc reduced to 
seconds add the constant quantity 4-6855749, and two thirds of the arith- 
metical complement of the log. cosine, the sum is the log, tangent of the 
given arc, 

PROBLEM VL 

To find a small arc from its log. sine or its log. tangent. 
10 



110 PLANE AND SPHERICAL TRIGONOMETRY. 

1, Let the log. sine be given ; then n being the number of seconds in 
the arc, the expression (2), in problem iv., gives 

log. n — log. sin. x — 4-6855749 -j- J arith. comp. log. cos. x 
= log. sin. ;?;-j-5'3144251 — lO-j-i arith. comp. log. cos. z; therefore, to 
find the arc from the log. sine the rule is this. To the log. sine of the 
small arc add 5'3144251, and \ of the arithmetical complement of the 
log. cosine ; subtract 10 from the index of the sum, and the remainder 
will be the logarithm of the number of seconds inthe arc. 

2. Let the log. tangent be given ; then from the expression (3), last 
problem, we have 

log, TO = log. tan. X — 4'6855749 — I arith. comp. log. cos. x 

— log. tan. X -\r 5-3144251 — 10 — % arith. comp. log. cos. x\ 
that is, to the log. tangent of the small arc add 5'3144251, and from the 
sum subtract | of the arithmetical complement of the log. cosine, take 
10 from the index of the remainder, and we shall have the logarithm 
of the number of seconds in the arc. 

Let us now apply each of the foregoing rules to an example. 

1. Required the log. sine of 1' 4-8754''. 

By the rule in problem iv. the process is as follows : 

log. 64-8754 - - - 1-8120801 

Constant No. - - 4-6855749 



6-4976550 
I arith. comp. log. cos. - - 



log. sin. r 4-8754'' - - - 6-4976550. 
By the tables the log. sine is found as follows : 

log. sin. 1' 5" - - - 6-4984882 

log. sin. 1' 4" - 6-4917548 

Difference - -0067334 

.-. log. sin. 1' 4-8754" = 6-4917548 + -8754 X '0067334 = 64976489. 

2. Required the log. tangent of 7' 2-38' 

By the Rule in Problem V. 
log. 422-38 - 2-6257033 
Constant No. 4-6855749 
I arith. comp. log. cos. 



By the tables. 

log. tan. 7' 3" 7-3119158 

log. tan. 7 2 7-3108879 



•0010279 



log. tan. 7' 2-38" 73112782. 

.-. log. tan. 7' 2-38" = 7-3108879+ -38 X -0010279 = 7-3112785. 

3. Required the arc whose log. sine is 6-4976550. 
By the Rule, Prohlem V. log. sine . 6-4976550 

Constant No. . 5-3144251 

I arith. comp. . 

log. 64-8754 . . 1-8120801 

,-. the arc is 1' 4-8754". 
By the Tables. 
The proposed log. sine lies between log. sine 1'4" and log. sine 1'5", 
and the difference between these logs is -0067334 ; also the difierence 
between the proposed log. and log. sine r4" is 59002; hence 

required arc = 1' 4" + -^^ = 1' 4-876". 



SURFACE OF A SPHERICAL TRIANGLE. lit 

4. Required the arc whose log, tangent is 7" 1644398, 
By the Rule. log. tan. . , 7-1644398 

Constant No, . . 5-3144251 

I arith. comp. . . — 3 



log. 301-2067 . . 2-4788646 

.-, the arc is b' 1-2067''. 

B]i the Tables. 
The proposed log. is between log. tan. W V and log. tan. 5' 2'' ; the 
difference of these logs, is -0014404, and the difference of the proposed 
and log. tan. 5' V is -0002981. 

2981 
.-. the arc is W V + -^ = 5' 1-2069''. 
' 14404 



CHAPTER II. 

INVESTIGATIONS OF EXPRESSIONS FOR THE SURFACE OF A SPHERICAL. 
TRIANGLE AND FOR THE SPHERICAL EXCESS. 

(84.) It has been already shown (36) that two great circles alv/ays 
intersect in two points at the distance of a semicircle from each other. 
The space thus included by two great circles is called a lune^ (see the 
fig. at p. 42.) 

The surface of a lune is to the surface of the whole sphere as the 
arc GIQ,', or as the angle P of the lune, is to the whole circumference 
IGlHI. This is pretty obvious, but it may be rigorously proved in the 
same way as it is proved in plane Geometry, that in the same circle 
any sector is to the whole circle as its arc is to the circumference, 
(Geom. prob. 23. Book 6). Hence, if we call the surface of the sphere 
S, and the angle of the lune w degrees, the expression from its area will 

be S ;-— - ; or if instead of degrees, co represents the absolute length of 
o60 

those degrees to radius 1, then the expression may be written S — , 
where t: stands for the number 3-14159, &c. ^'^ 

It can be proved, although not by the elementary principles of Tri- 
gonometry, that the surface of a sphere is equal to four times the area 
of one of its great circles;* that is, r being the radius of the sphere 
S = 4 -nr-, so that the expression for the area of the lune is 2r-w. If we 
suppose r to be unity, the surface will be expressed by 2aj, that of the 
whole sphere being 47r. 

PROBLEM I. 

To express the area of a spherical triangle in terms of its three angles. 

Let ABC be any spherical triangle, and produce the sides AC, BC, 

till they meet again in C, forming the lune CC. The j^ B 

triangle CAB will be a portion of an opposite lune equal 
to the lune CC ; and this portion will obviously be equal 
to the portion C'A'B', provided the arcs CA, CB, are 
equal to the arcs CA', CB'. Now AA' is equal to CC, ' 
each being a semicircle ; hence, taking from each the 
common part CA', we have CA = CA'. In like man- 
ner CB — CB', and, therefore, the triangles ABC, 
A'B'C, are equal. Hence the surface of the hemisphere, 
whose base is AB'A'B', is equal to the sum of the three 
lunes AA', BB', CC, minus twice the triangle ABC ; 

* See " The Elements of the Integi-al Calculus," page 144. 




112 TRIGONOMETRICAL INQUIRIES. 

that is, calling the surface of this triangle S. 
jS=2r2CA + B + C) — 2S 

.-. S = 7-2 (A 4- B+ C) — i S = ?-2 J(A + B -{- C) — :r j. 
where it must be observed that A, B, C, denote the lengths of the arcs 
which measure the angles of the proposed triangle to radius unity. 
But, if we take A, B, C, and n in degrees, then since 

180° :7r :: A-f B + C — 180°: JA + B + C — 180oj^- 
the expressionforS willbeS^rSjA+B+C— 180oj-^- . . (1). 

If the radius of the sphere, on which the triangle is, be taken for unity^ 
then calling the area in this case e, we have 

£ = A + B + C— 180° .... (2); 
which indicates that the area of a triangle, on the surface of a sphere^ 
whose radius is unity, is equal to the excess of its three angles above two 
right-angles. This quantity is technically called the spherical excess, and 
the theorem (2) is known by the name of Girard^s theorem. 

It follows from this proposition that two spherical triangles are equal 
in surface, if the angles of the one are severally equal to those of the 
other, or, indeed, if the sum of the angles of the one triangle is equal to 
the sum of the angles of the other. 

PROBLEM n. 

To express the area of a spherical triangle, or the spherical excess in 
terms of two sides, and the included angle. 

Calling as before the surface of the triangle to radius unity e, and, the 
sum of its three sides s. We have, by last problem, 

£ = 5— 180° .-. cot. •^ = — tan. 1 5. 

and, by Napier's analogy, tan. | (A + B) = — ' ^ ^ .J^ cot. | C; 

hence, by substitution, 

1 __ ^°^- 2 C*^ — *) cot. I C -{- COS. i{a-}-b) tan, j G ^ 
'2 COS. ^ (ft — b) — COS. |(a-}-6) ' 

or multiplying the numerator by 2 sin. |C cos. gC, and the denominator 
by its equal, sin. C, (equa. 18, p. 37,) 

€ _ 2 cos. i (a — ^)cos^HC_+2_cos. h (a + ^>) sin.^ ^ C . 
' 2 ~ cos. i {a — b) sin. C — cos. h{a-\-b) sin C 
that is, by the formulas (1) and (2), page 32, 

£ cos. ^ a cos. hb-\- sin, h cl sin. \ b (cos.^ | C — sin.^ | C) ^ 
2 "" sin. I « sin. I b sin. C ' 

but, (from 19, p. 37,) cos.^ i C — sin.^ IG = cos. C ; hence 

, £ cot. I a cot. i & + cos. C ( cot. | a cot. 1 5 , , > ^ _, 

cot. — zr: 5 _2_^' ^ J ^ ^ 1 \ cot. C. 

2 sm. C i cos. C 3 

To adapt this expression to logarithmic computation suppose first that 

„. . . , , cot. J a cot, ^b 

COS. Cis positive, and that we assume ;;; = tan. ^ e, 

COS. G 

then cot, le — sec. ^ o cot. C ; suppose, secondly, that cos. C is negative, 

^, ..cot, i« cot, lb . . „ , 1 -I- • 1 

then if —7T~~ IS numerically less than radius, assume it equal 

cos. Kj 



SPHERICAL EXCESS. 



113 



to sin. 2 d, and we shall have cot. — - = cos. 2 cot. C ; but if the same 

expression be numerically greater than radius, then assume it equal to 

sec. 8 e, when we shall have cot. -^ = tan. 2 cot. C. . 

It may be remarked that, with the proposed data, the excess may be 
otherwise easily determined, by iirst finding, by the common formula, 
the third angle of the triangle, and then applying Girard's theorem^ 

PROBLEM m. 

To determine the spherical excess when the three sides are given. 
By formula 25, p. 38, 

1 -f COS. a 1 -\- cos, b l-j-cos. a-{-cos. b-\-cos.acos.b 



cot. i a cot. ib=- 



sin. a ' sin. b sin. a sin. b 



•D„ ^^^^ ,1 /AX A^ ^ COS. C — COS. « COS. 6 

By formula (A) p. 47, cos. C =- 



sin. a sin b 
By formulas (1), (2), p. 49 

o 

2sin.jCcos.|C==sin.C=^-T^^— -^^4Ssin.(iS-«)sin.(|S-6)sin.aS— c). 

Substituting these values in the expression for cot, ~, last problem, 

we have 

1 -|- COS. a -\- cos. b -j- COS. c 



cot. 



2 



2 Vsin. I S sin. (i S — a) sin. (i S — J) sin. (* S — c) 
,. , , ... ^^ 



. (2). 



We may investigate another expression for the excess, as foUowa; 
By the formulas (1), (2), page 49, 

sin.§Acos.^B=.'?^-^^^~^^ | sin.|Ssin.(§g^) 
sin. c J g^j^_ f) gjjj_ g 

sin.§Bcos.iA=:'i^H^^^ | sV J S sin. Q S -~c) 



By adding, sm. h (A + B) == ■ — \^ . cos. J a 

^ 2 sm. J c cos. I c 

rs 1 • • , « •r.x sin. (A S — b) — sin. (i S — a) , _ 
By subtractmg, sm. I (A — B) = — ^' . , ^^ — ^ cos. ^ C. 

^ &j a V >» 2 sm. h cos. I c 

But by formula (27), page 39, 

sin, (i S — b') -\- sin. (§ S — a) =2 sin. | c cos. J (a — &). 

sin. (i S — b) — sin. (^ S — a) = 2 cos. J c sin. | (a — 6). 

__ , 1 . . . , . I T, cos. 5 (a — b^ _, 

Hence by substitution, sin. J (A + B = ^ ^ cos. * C 

COS. i c 

sm; i (A — B = ^^^^ — ^ cos. I C. 

sm. ^ c 

Proceeding in the same way with the expressions for cos. | A cos. | B, 
and sin. | A sin. | B, 

VAK sin. h (a -^ b) . „ 

there results cos. K A — B) = f— ^ — ! — ^sin. J C, 

^ ^ sm. I c ' 

cos, K A + B) = £^!iii^±i2 sin. J C. Now, since 
10* '0 



114 TRIGONOMETRICAL INaUIRIES. 



siii.4-=— cos.l (A-j-B+C) =sin.i (A+B) sin. I C— cos.A (A-fB) cos.| C, 

we have, by substituting in the second member the first and last of the 

n . . . £ sin.iasin.i^» . ^ .-. .• ^ 

loregomg expressions, sm. — = ^ sm. C ; or substitutmg for 

^ COS. 2 ^ 

sin. C its value, as exhibited in last problem, and recollecting that (31), 
sin. a sin. b = ^ sin. J a cos. J a sin. J b cos. i b, 

. £ 1 sin. i S sin. (i S — a) sin. (i S — 6)sin. aS — c) ,_, 

we have sm.-— =: '^ ^— — ■ = (3) ; 

2 2 COS. J a cos. | b cos. I c 

an expression adapted to logarithms. 

By combining the formulas (2) and (3) various others may be deduced . 
Thus, by multiplying them together, we have 

£ l + cos. « + cos. 5 + cos. c ,.. 

2 4 cos. i a COS. | b cos. ^c ' ' ' 

But, formula (20), page 37, 
cos. I a z= V (i 4" a COS. a), cos. J 6 = V (i + |cos.Z»),cos.|c=:: Vd-f-aCOS.c) ; 

- e 1-1- COS. a 4- COS. b -\- cos. c ,_. 
hence, cos.-—- = — ' - .... (5). 

^ ^2(l + cos.a)(l + cos.6)(l-|-cos.c) 

_, • .-.- o o e (1-f-cos. a-f-cos.^>-f-cos.c)2 

Squarmg this, 2 cos.^ -— = ; --—^ ' — <- — r- ; 

^ * ' 2 (l-|-cos.«)(l-fcos. ^>)(l-|-cos. c) ' 

which, since cos. £ = 2 cos.^ -- — 1, gives 

_ (l-|-cos.a-|-cos.J4"COS.c)2 — (l-f-cos.a)(l-|-cos.5)(l-{-cos.c) .^. ^ 
cos.e- (r-l-cos.«)(l-|-cos.*)(l-|-cos.c) • • W; 

also because 1 — cos. c = vers. «, we may change this into 

1 — cos.2 a — C0S.2 & — cos.2 c -f- 2 cos, « cos. b cos. c 

vers. £ = - ' — 

(1-j- COS. a) (1 -{- cos. Z*) (1 -j- COS. c) 

Lastly, by squaring the expression (3) and multiplying by 2, we have 

- . „ £ , sin.iSsin.(iS— a)sin.(iS— Z»)sin.(|S— c) ... 
2sin.2-Tr-=l— cos.£=vers.£= .r — ^^-^ -„ . (8). 

2 2 C0S.2 i <J C0S.2 1 b COS.2 i c '^ ^ 

The expression, marked (2), is due to De Gua, as are those marked 
(4), (5), (6), and (7). The expression (3) is from Cagnoli, (Trigon. page 

£ J COS. ^ £ 

329.") Since, tan.—;- = : '-^—^ we have, by combining the expres- 

4 sm. ^£ or 

sions (3) and (5), 

£ 1 — COS. ^^a — COS.- ib — C0S.2 he 4-2 cos. | a cos. J i cos. | c 
tan. — =; ■ — 

^ "^ sin. i S sin. (i S — a) sin. ihS — b) sin. (i S — c) 

Now, 1 — C0S.2 J ft — C0S.2 J ^ = sin.2 i a — cos.^ h b, 

by equa. 5, p. 32, = sin.^ ^ a sin.^ i J — cos.^ i a cos.^ i h ; 
hence, the numerator of the above expression is equal to 
sin.2 A a sin.2 hb — (cos. J a cos. 1 6 — cos. J c)^, which is the same as 
\ sin. J « sin. ^ Z) -|- cos. J « cos. J Z> — cos. J c | X 
^sin. |«sin. |Z> — cos. | a cos. 1 6 -{-cos. J c^ ; or as 
Jcos.K«— ^)— cos.JcJXjcos. Jc — COS. ^(ft-|-Z')J = (byequa.27,p.39); 
2 sin. ^(^S — Z>)sin.J(|S — «)X2sin. iSsin.lQS — c). 

Consequently, since (page 38), V | tan. | A = ^^^' " 

g ■^ sin. A ' 

the foregoing expressions for tan.— takes this very remarkable form, viz. 



(7). 



SPHERICAL EXCESS. 115 

tan, -^ = Vtan.i Stan, k (iS — a) tan. ^ (a S — Z')tan.| (i S — c); 

which is Lhullier's expression. 

It follows from this problem that two spherical triangles are always 
equal in surface when the sides of the one are severally equal to those 
of the other, whether the triangles admit of coincidence or not. 

PROBLEM IV. 

Given the area of a spherical triangle on the surface of the earth in 
square feet, to determine the spherical excess. 
Let the area of the triangle in feet be S, then, by problem i., 
_ S_ ISQo 

'~r2"' 314159' 

Now the length of a degree, supposing the earth to be a perfect sphere, 

is 365154'6 feet; hence the earth's radius is ^ — X 365154*6 feet j 

314159 S ^ ... , 

consequently, e = ^^^ .nrr^rAn^ degrees; or if the excess be expressed 
loU X {ooDiO'i'op 
J ^ 3-14159 SX 602 

in seconds, then e = _^-^^^^_ seconds. 

.-. log. e = log. S -f log. 62-83185 &c. — 2 log. 365154-6 
= log. S + 1-7981799 — 11-1249536 
= log.E— 9-3267737. 

Hence, from the logarithm of the area of the triangle in feet, subtract the 
constant logarithm 9-3267737, and the remaind^er loill be the logarithm, of 
the excess in seconds. 

This rule, which usually goes by the name of General Roi/s rule, is 
in fact due to the late professor Dalby, by whom it was communicated 
to the General, when engaged with him in conducting the Trigonome- 
trical Survey. (See the " Life of Mr. Dalby," in Leybourn's Reposi- 
tory, vol. V.) 

By means of the rule just given we may very readily compute the 
spherical excess, provided that we previously know the area of the tri- 
angle in feet. In trigonometrical surveying, the triangle on the surface 
of the earth, composed between any three stations, is necessarily so li- 
mited a portion of the whole sphere that its area, computed as a plane 
triangle from the measured data, cannot be affected with any error of 
consequence. On this hypothesis, therefore, the area of the triangle 
may be determined by one or other of the methods in prob. ni., last 
chapter, and thence the excess ascertained by the above rule. Should 
the excess, thus deduced, exactly equal the excess of the three observed 
angles above two right-angles, we may be assured of the accuracy of 
the observations ; but if they differ, the difference must be regarded as 
the amount of the errors with which the three observed angles are 
affected. If all of them were observed with equal care, so that there 
appear no reason why one should be more erroneous than another, the 
correction thus found must be distributed equally among them ; but if it 
be suspected that one of the angles is less to be depended on than the 
others, then to this angle must be applied the greater part of the whole 
correction. The data being thus corrected, the required side or sides 
of the spherical triangle may be computed by the rules of spherical tri- 
gonometry; or the same object may be effected by plane trigonometry, 
with all requisite accuracy, provided we employ in the computation, 
not the corrected spherical angles, but these angles diminished each by 
one third of the spherical excess found as above, a truth which has been 
established by Legendre, (See the Appendix to Brewster's translation 



116 TRIGOXOMETRICAL INaUIRIES. 

of Legendre's Geometrj'.) Trigonometrical surveying is a very im- 
portant application of the theorj- of trigonometry', but" is foo ample a sub- 
ject to admit of being discussed in the present volume. The student 
will find a condensed account of these geodetical operations in the tenth 
section of Dr. Lardner'"s Trigonometry, and every requisite information 
in the Geodesie of M. Puissant and Col. iViw^g-e's account of the Trigo- 
nometrical Survey of Engla.7id and Wales* 

Miscellaneous Expressions involving the Sides o/nd. Angles of a Spherical 
Triangle. 

(85). "We shall terminate the present chapter by the insertion of a 
few general expressions, invohong the three sides and the three angles 
of a spherical triangle. Those formulas which have already been 
given in the second part of the work, are amply sufficient for the so- 
lution of every case in spherical trigonometry, but the sides and angles 
of a sphericartriangle possess many other remarkable relations whicli 
are often called in aid, in higher investigations concerning a sphere. 
A few of these, therefore, it maybe proper to give. Lets represent 
half the sum of the sides a, Z>, c, and S, half the sum of the angles A, 
B, C, of a spherical triangle : then by multiph'ing together the expres- 
sions for sin. *A, cos. J A^in art. (47)', and those for sin. \a, cos. Ja, in 
art. (49); and squaring the results, we have these equations ; 
sin. -5 sin. ^c sin. ^A = 4 sin; s sin. (s — «) sin. (s — V) sin. {s — c)== 471^. (l) 
sin. 2B sin. ^C sin. ''-a = — 4 cos. S cos. (S — A) cos. (S — B) cos. (S— C) 
= 4N-2 (2). 

By multiplication, -■. sin. a sin. h sin. c sin. A sin. B sin. C = 4 Nti (3). 

„ T. . . sin. b sin. c sin. A n 

By division, -^ — ^ . . ^ : -. = -^ . 

•^ sm.B sm.C sm. a N 

But the first two factors of this expression are each of them the re- 
, ^ , , sin. 6 sin. c sin, a ti 

ciprocal of the last. .-. -. — - = —r—^ = -. — r- = -:r=r . . . . (4). 
^ sm.B sm.C sm.A N 

„ ^ ^,^ sin.A 2n N 2n 

But from (I),-. = -. — r- •■• — = -^ : — r-^ • (4). 

sm. « sm. a sm. sm. c 7i sm: a sm. o sin. c 

and from (2), 

sin. a 2N n 2N 



(6). 



sin.A sin. A sin. B sin, C ' N sin. A sin. B sin. C 

Substituting in (6) the value of N deduced from (5), and in (5) the 
value of n deduced from (6), we have, from the resulting equations, 
these expressions for n and ^nT viz. ^ 

% =r i 5 sin. 2 a sin, ~b sin. -c sin. A sin. B sin. C p . : . (7) 
N = i jsin. 2Asin.2B sin.-^Csin. a sin. 6sin. c|* . . : (8); 

and, for their ratio — , we have from these, as also from (4), 
n sin. a sin. b sin. c , 

"N "^ ^ilTAsin.Bsin.Cr ' ' ' ^^^5 
expressions which are remarkable for their sjonmetry. 

Again, referring to the expressions for sin."|A, and cos. |A, at (47), 

72,2 

we see that sin, h A sin, | B sin, | C = -. : — j—. (10) 

sm. 5 sm. a sm. b sm. c "^ 

, A ■r> , ^ "^ sin. s .,,^ 

cos I A cos. J B COS. * C =^ -. — -—. — . . . (11) 

sm. a sin. b sm. c 

* Some additional particulars respecting the spherical excess will be found in the 
gupplenient. 



MISCELLANEOUS FORMULAS. 117 

.*. tan. i A tan. ^ B tan. I C = -r^ .... (12). 
sm.2 s '' 

And by referring to the expressions for sin. |«, cos. |«, at (49), we 
see the truth of the following analogous equations ; viz., 

. , . , t . , — Ncos. S ,,„^ 

sm.iasm.^osm.ic = — — r — : — =—-. — — -* . . (13) 

sin. A sm. B sm. C ^ ^ 

COS. i a cos, 1 6 COS. | c = ——. — -—. — — -^ — — . (14) 

— cos.Ssm. Asm.Bsm.C ^^ 

.-. tan. J a tan. ^ J tan. ^ c = — ^-— . . (15). 

ttP- 
Prom (10), sin. s = -: ^ — — : : —-. — -, . (15) 

sin. a sm. b sm. c sm. J A sm. | B sm, J C ^ 

= '''y^^>' asin.iAsiriBsin.iC ' ' ^^^^^ 

_ ,,,. . sin. a sin, 5 sin. c cos. * A cos. JB cos. iC 

From (11), sm. s = = 5_ 

n 

= ,by(5),2^cos.jAcos.iBcos.iC . . (17); 

and, from (12), sin.2 5 = ^^- — - . . (18). 

' ^ ^' tan. i A tan. J B tan. i C ^ 

In like manner, from (13), (14), (15), we have 

„ sin. A sin. B sin. C sin. i a sin. \ h sin. h c 
COS. S = ^ ? H_ 

N 
= ,by(6), — 2 — sin. J « sin. I J sin. J c . . (19) 

cos. S= — — 



sin. A sin. B sin. C cos. J a cos. ^ b cos. I c 

' ^ ^ ^' 2 cos. I a COS. i 6 cos. I c 

N 

C0S.2 S = N tan. ^ a tan. | b tan. | c = ; . (21). 

cot. I a cot. A 6 cot. J c 

(86). In addition to these we shall here put down a few other useful 
expressions immediately deducible from the four equations which we 
had occasion to investigate at p. 114 ; and which are as follows : 

sin.KA + B)=:-^^^^*-^cos.H« — ^') . . . (22) 
COS. i c 

sin.HA — B) = -^?^:l^sin. ^(«-*) . : . (23) 
sm. I c 

cos.§(A + B)=: ^'^-^^ cos.h{a + b) . . . (24 

sin *■ C 
COS. KA— B) = .'\ - sin. l{a-^h) . . . (25). 

From these equations we immediately deduce the following analo- 

* This expression, as well as those marked 19, is usually given with an 
improper sign, viz. -\- instead of — , a mistake which seems to have 
arisen from confounding V{cos. S ' cos. S) with V{ — cos. S X — cos. S,) 
which are, in fact, distinct expressions ; the one being -J- cos. S, and 
the other — cos. S. See the chapter on Imaginary Quantities, in 
Young's Algebra, just published by Carey, Lea, & Co. Philadelphia. 



118 ASTRONOMICAL PROBLEMS. 

gous ones, viz. sin. i{a-j-b)= ^^^ cos. KA — B) . (26) 

sin. gO 

sin.H«-*) = ^^sin.HA-B) . . . (27) 

cos.H« + ^)=^?^cos.i(A + B) . . . (28) 
sm. 2 '-' 

cos.i(«-^') = ^^'^sin.|(A + B) . . , (29). 

From (22) and (2.3) we have, 

sin.2 HA -f- B) C0S.2 1 c = cos.2 a C cos.2 ^(a — b) 
sin.2 i (A — B) sin.2 1 ^ ^ cos.2 | C sin.2 i {a — b). 
Hence, by addition, 
sin.2^(A — B) sin.2 ic-fsin.2i (A + B) cos.2 ^c = cos.HC _ _ ^30^^ 

In like manner, from (24) and (25/, 
cos.2 HA — B) sin. 2 1 c + cos.2 i (A -f- B/ cos.2 j ^ ^ sin.2 ^ C . . (31). 
Again, from (26) and (27), we have 

sin.2 i (a -j- 6) sin.2 i C = gin,?, i ^ cos.2 1 (A — B) 
sin.2 iia — b) cos.2 xc= sin.2 1 ^ sin.2 1 (a — B) ; 
and, by addition, 

sin.2 Ha — *) cos.2 J C + sin.2 ^{a-\-b) sin.2 J C = sin.^ J c . . . (32) ; 
and, in like manner, from (28) and (29) we get 
cos.2 i(^a—b) cos.2 i c _j_ cos.2 j (« + i) sin.2 x c = cos.2 j c . . . (33). 



CHAPTER III. 

ON THE RELATIONS BETWEEN THE CORRESPONDING VARIATIONS OP THE 
PARTS OF A TRIANGLE. 

In the present chapter we propose briefly to examine into the effect 
produced on the sides and angles of a triangle, by a small change taking 
place in the magnitude of one of them ; that is to estimate the amounl 
of error affecting any part which may have been determined from data, 
not strictly accurate, and thence to ascertain under what circumstances 
a small inaccuracy in a proposed datum will least affect the accuracy of 
the result. This becomes a very essential matter of inquiry in all the 
more delicate practical operations of trigonometry, because, since the 
data furnished by observation necessarily fall short of strict accuracy, 
on account of the imperfections of instruments, and other unavoidable 
defects, we ought to know under what circumstances our observation 
should be made, so that the small error with which it is affected may 
have the least possible influence on the quantity to be determined from 
it. The following problems will sufiiciently show the method of arriv- 
ing at this knowledge. 

PROBLEM I. 

In a right-angled plane triangle, whose base is &, and altitude a, it is 
required to determine the error committed in calculating a by means 
of the given base b, and the observed angle opposite to a. 

Let us consider a to represent the true angle opposite a, from which 
that given by observation varies by a small quantity, which we shall 
represent by 'Sa, and call the variation of a, then the sought side which 
would be given by the equation a=.b tan. a, is affected by an error 6a, 



MINUTE VARIATIONS. 119 

SO that instead of a it is a -j- ha, and this we determine from the equation 
a-\- ha-=h tan. (a -\- <5a) ; in which, by subtracting the preceding equa- 
tion, we find the value of ba, to be, 

la=h\ tan. (a -f Ja) — tan. a X = ' "', ^ ^ (art. 27): 

^ ^ » COS. a cos. (a-|-(5a) ^ / 

Now, by hj'pothesis, 6a is very small, so that we may substitute it for 

its sine, and cos. a instead of cos. (a -\-6a), .-. 6a = — : 

' cos. 2 a 

m which expression Sa is the length of the arc to radius 1, which mea- 
sures the angular error. 

To determine what length b must be to render the variation ,0a the 
least possible under the same amount of error 6a in a, we have 
acoi. a6a a6a ^ aha 

b = a cot. (z .•. da = • == — = 2 — : — - — ; 

COS."^ a sm. a COS. a sm. 2a 

hence 6a will be the least possible when sin. 2a is the greatest possible, 
that is when a =: 45° : so that in order to determine the height of a 
tower or steeple, &c, with the utmost accuracy, by means of an obser- 
vation of its angular altitude, we should make the observation at a 
distance from the object as nearly as possible equal to its height. 

PROBLEM n. 

In a right-angled spherical triangle is given one of the oblique angles 
to determine the variation of the opposite side, arising from a small 
variation of the hypotenuse. 

Let A be the constant angle, a its opposite side, and c the hypotenuse; 
then sin. a = sin. A sin. c, sin. {a -j- ha) = sin. A sin. (c -|- he) 
.'. by subtraction, sin. {a-{- 6a) — sin. « = sin. A ^sin. (c-j-Jc) — sin.c^; 
that is, (page 39, equa. 27,) 

2 cos. (<^ -f- 5 6a) sin. | J« = 2 sin. A cos. (c -j- 2 ^c) sin. h he 
. , ^ sin. A cos. (c -j- i ^c) sin. ^ (5c j •-. , , . n 

.*. sm. ^ 6a = 7 — V^ — ~ ; and if ha, he, be very small, 

COS. {a-{- i ha) ' ' ' ^ ' 

sin j^ cos c 

ha = — '- ^— (Jc: or, substituting for sin. A its value from the first 

COS. a 

sin. a COS. c .■,■■,•■•■,■, 

equation, 6a = — 6e = tan. a cot, e 6e: which variation will 

COS. a sm. c 
be the least possible when cot. e is least, or when c = 90°. It would 
seem from the expression for ha, that in this case ha is absolutely 0, which 
we know cannot be. Indeed, no result deduced like that above, from a 
process in which certain small quantities are rejected, can be considered 
as perfectly accurate, although they may approximate so nearly to the 
truth as to be practically admissible as such. If we restore the ihc which 
has been neglected, and write the above result thus, ha — tan. a cot. (c 
4- i 6c) 6c; then, in the case of c =r 90°, the expression becomes ha = — 
tan. a tan. i he • he; or, considering the very small arc JJc to be equal to 
its tangent, we have in the case supposed ha= — ^ tan. a {he)'^, the same 
expressions otherwise determined by Professor Airy in his Treatise on 
Trigonometry, in the Encyclopaedia Metropolitana, 



PROBLEM in; 

In an oblique-angled spherical triangle are given two sides to deter- 
mine the variation produced in the third side by a small variation of the 
opposite angle. 

Let a, b,he the two given sides, G the included angle, and c the side 



120 TRIGONOMETRICAL INQUIRIES. 

opposite to it. Then cos. c = cos. a cos. b -\- sin. a sin. h cos. C, 

COS. (c -\- Sc) = cos. a cos. b -f- sin. a sin. 6 cos. (C -f- Jc) ; 

.-. by subtraction, cos. (c -f- Jc) — cos. c = sin. a sin. 6 ^ (C -f-^C) — cos. C^ • 

that is, 2 sin. (c -[- i ^c) sin. i Sc =2 sin. a sin. b sin. (C -[- J JC) sin. J ^C 
Hence, if JC be very small, sin. c 6c = sin. a sin. 6 sin. C(5G 

sin. a sin. ^» sin. C .^ . • -d ,/>. 

.-. Sc = ^ SC = sm. a sin. BJC ; 

sm. c 
and Sc is therefore the least possible when sin. C is the least possible, 
that is, when C =0. To find the expression for Sc, in this case, restore 
what has been rejected, and we shall have 

sin. a sin. b sin. (C + * J C) ,^ , . , , ^ „ -, .« 

Sc = : — ^ ^— ^— SC : which, when C = 0, and | SC 

sin. C > a 

very small, becomes Sc = —^r—. — •• — (SCy. 
•' ' 2 sm. c ^ 

PROBLEM IV. 

In an oblique-angled spherical triangle are given, as before, two sides 
and the included angle, to find the variation produced in one of the 
opposite angles by a small variation in the included angle. 

Let a, b, be the given sides, C the included angle, then we have to 
find what influence a small variation in the value of the angle C will 
have on the angle A opposite a. For this purpose we shall deduce a 
suitable formula, as follows : substitute the expression for cos. c, on the 
opposite page, in the corresponding expression for cos. a, and we shall 
have the equation cos. A sin c = cos. a sin. b — sin. a cos. b cos. C ; 
» sin. c . , , ^ ^ sin. c sin. C 

.'. cos. A -; — — = cot. a sm. o — cos. b cos. C. But -: = ~ — r ; 

sin. a sm. a sm. A 

hence by substitution, cot. A sin. C = cot. a sin. b — cos. b cos. C, 
cot. (A + ^ A) sin. (C + ^C) = cot. a sin. b — cos. b — cos. (C + SC) ; 
and by subtraction, 
cot.(A + S A) sin.(C + SC) — cot'.Asin.C = cos.^» Jcos.C— cos.(CH-^C)|. 

The first side of this equation is the same as 
cot.(A + <?A)Jsin. (C + JC) — sin.C J + sin.C Jcot.(A+^A)— cot. A^ ; 
and the quantities within the brackets are respectively the same as 

2 cos. (C + i (5 C) sin. i SC and — : 7" ^^^'^ ^ , ^an 

^ ' ^ sm. Asm. (A+JA) 

Also the second side of the same equation is the same as 
cos. b • 2 sin. (C -f- 2 <5C) sin. i SC ; consequently, 
2 CO., (A + .A) COS. (C + i .0 s:n. J .C - -J^.^^ = 
2 cos. b sin. (C -j- | SC) sin. ^ SC ; and, therefore, when SC and SA are 

very small, cot. A cos; CJC — . '., . ^A = cos. b sin. CJC 
^ sm.^A 

.-. JA = -T-'—^ (cot. A COS. C — COS. b sin. C) ^C. 
sin. C 
The foregoing examples are those selected by Professor Airy in his 
Treatise on Trigonometry, before referred to, and we have here adopted 
his processes. But the instruments of investigation generally the best 
adapted to inquiries of this kind is the Differential Calculus. 



END OF TRIGONOMETRY. 



SUPPLEMENT 

ON SPHERICAL GEOMETRY, POLAR TRIANGLES, &c, 

BY T. S. DAVIES F. R. S. E., F. R. A. S. &C. 



CHAPTER I. 

ON SPHERICAL GEOMETRY. 

In the commencement of the Spherical Trigonometry, a small col- 
lection of propositions, such as were necessary in the character of fun- 
damental principles upon which to build the subsequent analytical 
investigations was given. At the request of the author, we here pro- 
pose to add a few others, and shall endeavour to select such as may 
serve the double purpose of facilitating our future inquiries, and of 
interesting the mind of the student in some of the most beautiful classes 
of Geometrical research that are yet known to exist; we shall com- 
mence with a few properties analogous to the more elementary propo- 
sitions in Euclid, and which are very often assumed by writers in spne- 
rical trigonometry, both unnecessarily and improperly. 

1. Let O be the spherical centre of a circle, 
and AB any great circle chord : the perpendicu- 
lar* OK demitted from the centre upon AB will 
bisect it. Draw AO, BO. Then from the right- 
angled triangles AKO, BKO, we have. 

cos. OA _ COS. AK cos. OK cos.' AK 

cos. OB ^ COS. BK cos. OK ^ cos. BK ' 
But OA = OB, and .-. AK = BK. 

2, Conversely, if OK bisect AB, it will cut it 

at right angles. 

^ » ^^^ cos. AO — COS. AK cos. KG 

For COS. AKO = - 




COS. BKO 



sin. AK sin. KO 
COS. BO — COS. BK cos. KO 



sin.BKsin. KO 

But the right-hand sides of these equations are equal, term for term, and 
therefore cos. AKO = cos. BKO, or AKO == BKO; and as AK, KB 
are one great circle, the angles at K are right angles : the tenth defini- 
tion of the first book of Euclid applying to spherical as well as to plane 
angles. 

3. If the great circle chord AB, be bisected at right angles at K, by 
the great circle ZM, this perpendicular shall pass through the centre 
of the circle. For, assume for a moment that the centre is at O^ not 
in the circle ZM ; and draw the perpendicular O'K'. Now, we have 
seen that O'K' bisects AB in K' when O' is the centre, or that AK -f- 
KK' = BK — KK' ; But, by hypothesis, AK = BK, and therefore, sub- 
tracting the latter equation, KK' = — KK', which is only true when 
K, K' coincide, that is, when O'K' coincides with OK, or when O' is 
in ZM. The centre is therefore in ZM. 

» Always meaning a great circular perpendicular, except e.xpressly stated otherwise. 
II P 



122 



SPHERICAL GEOMETBT. 




4. If two great circles which cut one 
another at A, be intersected by a circle of the 
sphere in D, E, and H, L respectively, the 
rectangles of the tangents of the semi-seg- 
ments into which they are divided shall be 
equal. 

That is, 
tan. i AE tan. | AD = tan. ^ AH tan. i AL. 

For find the centre G, draw AG meeting 
the circle in B and C, draw the perpendicu- 
lars GF, GK, and join GD, GE, GH, GL. 

Then, cos. AK cos. KG = cos. AG = 
cos. AF cos. FG, cos. LK cos. KG = cos.GL 
= COS. FE cos. FG. 

From these we have, by subtraction, addi- 
tion, and subsequent division. 
(cos. AK— C OS. KL) _ (cos AF — cos.FE) ^ 
(cos. AK -f7os. KL) ~ (cos. AF-j-cos. FE) ' 
and hence, by dividing (28) by (29), page 39. 

tan. KAK-KL) tan.^AK + KL) = tan.KAF — FE)tan. KAF-fFE), 
that is, tan. J AH tan. | AL = tan. l AD tan. i AE. 

The analogue to Euclid m. 35, maybe seen in another form in the 
Math. Repository, No. 23, part n., p. 131, 2. 

5. Let the secants in the case where A is without the circle take the 
position of tangents. Then D, E, F, coalesce, and so do H, K, L. 
Then the equation just obtained becomes 

tan.2 1 AF = tan.^ i AK, or AF = AK. 

The tangents from any points to the circle are 
therefore equal. The case when the point is 
within the circle is demonstrated by Cagnoli, 
in his Trigonometry, but the other case he has 
not noticed.* 

6. We may easily prove, also, that the great 
circle drawn through K at right angles to the ra- 
dius, OK, touches the circle. For draw any other arc from O, as OL. 
Then, because K is a right-angle, we have cos. LK cos. KO = cos. LO. 
But COS. LK < 1, and therefore cos. LO < cos. KO, or LO > KO ; and 
L will therefore fall vnthout the circle ; or, no part of K A falls within 
the circle whence KA is a tangent. 

It is unnecessary to dwell at greater length upon these simple sub- 
jects ; the nature of the inquiry, and the method of pursuing it, as well 
as its close analogy to the corresponding properties in the Elements of 
Euclid, must be at once apparent. 

We may add that in the Repository, as above referred to, some other 
remarkable analogies to plain properties are derived by similar methods, 
10 which we refer the inquiring reader. 

* Since this paper Avas written, Professor Lowry has sent me the enunciation of the 
proposition in the text, accompanied by the foregoing remark, and with the following 
corollaries subjoined, viz. 

1. If ail arc be drawn perpendicular to the diameter of a small circle of the sphere, 
the square of the tangent of half this arc will equal the product of the tangents of half 
the segnients into which it divides the diameter. 

2. If, from the extremities of the diameter of the small circle, arcs be drawn cutting 
the circle in the same point as the perpendicular, then the sum of the squares of the 
sines of half these arcs will equal the square of the sine of half the diameter of the 
small circle. 




SPHEKICAL GEOMETRY. 



133 





7. Let any spherical triangle be cut by a trans- 
versal cba. Then the products of the sines of the 
alternate segments will be equal. That is, 
sin. A.C sin. Ba sin. C^ == sin. cB sin. aC sin bA.. 

For sin. Ac : sin. Ah :: sin. / b : sin. / c 
sin. aB : sin. Be :; sin. / c : sin. / a 
sin. Cb : sin. Ca :: sin. / a : sin. / b. 
Hence, multiplying, and effacing the common 
terms in the antecedent and consequent of the re- 
sult, we have the properties stated above. 

8. The student can show, ex absurdo, that if the equality above stated 
takes place, the three points a, b, c, are in the same great circle. 

9. If through any point P on the surface of the 
sphere three great circles be described, which also 
pass through the angles of the triangle ABC, and 
cat the opposite sides in a, b, c, respectively, then 
sin. Ac sin. Ba sin. bC = sin. aC . sin. Be sin. bA. 

For the two spherical triangles BaA and CaA 
cut by the two transversals Cc, Bb, give respect- 
ively 

sin. AP • sin aB sin. bC = sin. «P sin. Be sin. bA, 
sin. aF • sin. cB sin. cA = sin. AP sin. aC sin. cB ; 
which multiplied, and the common terms effaced, give the enunciated 
property. c' 

10. If through any point P in a given 
great circle Aa, which passes through an 
angle, of a spherical triangle, great circles 
be drawn to the remaining angles cutting 
the opposite sides in b, c, respectively, then 
the great circle be will always pass through 
the same f)oint a' in the great circle BC, and 
so divide it that 

sin. Ba : sin. ac :: sin. Ba' : sin. a'C. 
For by (9 and 7) we have, respectively, 
sin. Ba : sin. aC :: sin. Be sin. Ab : 
sin, cA sin. bC 
sin. Ba' : sin. a'C :: sin. Be sin. Ab : 
sin. eA sin bC ; 
■when, by equality of ratios, we have' 

sin. Ba : sin. aC :: sin. Ba' : sin. a'C* 

11. If three great circles be drawn 
through the angles of a spherical triangle 

and through the same point on the surface of the sphere, cuttmg 
the sides in three points ; three other great circles, each passing through 
two of these points, will intersect the sides of the triangle (produced or 
not as the case may require,) in the circumference of one and the same 
great circle. 
By (10) we have sin. Ba : sin. aC :: sin. Ba' : sin. a'C 
sin. Cb : sin. bA :: sin. Cb' : sin. b'A 
sin. eA : sin. eB :: sin. e'A : sin. c'B ; 
or by multiplying vertically, and bearing (9) in mind, 

■* This division of an arc is analogous to that which in piano is called the harmonical 
division of a line. Some of the most interesting properties of elementary geometry 
flow from considerations respecting the mode of division ; and the spherical properties 
have perfect analogies to those. A few of these may be seen 4n the paper above men- 
tioned in the Repository ; and others will appear in a future number. Some curious 
investigations on this subject, by Professor Lowry, may also be seen in vol. n., new 
series of the same work, quest. 223. His processes however, are totally different from 
those just adverted to. 





124 SPHEEICAL GEOMETRY. 

sin. Ba' • sin. Cb' sin. c'A = sin. «'C • sin. b'A ' sin. c'B ; 
and hence by (8) the prSposition is established,* 

12. If great circles be drawn from the angular points of any spherical 
polygon to a point on the surface of the sphere, the product of the sines 
of the alternate angles vrill be equal. In the triangle, (fig. to 9\ 
sin. BP : sin. PA :: sin. BAP : sin. ABP 
sin. PA : sin. PC :: sin. PC A : sin. PAC 

sin. PC : sin. PB :: sin. PBC : sin. PCB ; and, by mnltiplica- 
tion, sin. BAP sin. PCA ■ sin. CBP = sin. ABP sin. PAC sin. PCB. 
The student is required to prove it for four, fire, &c. sided figures, 
and is recommended to complete the argument from the suggestions 
furnished by the particular cases of the general truth. This theorem 
is due to Professor Lowry, Math. Rep. old series, vol. i., page 90. 

Let ilBC be a spherical triangle, and P 
a point on the surface of the sphere, from 
which perpendiculars PE, PF, PG, are 
drawn to the sides of the triangle : then the 
product of the cosines of the alternate seg- 
ments will be equal to one another. For b^ 

cos. AE cos. EP = cos. AP = cos. AF cos. FP 

cos, BG COS. GP = cos. BP = cos. BE cos. EP 

cos. CF COS. FP = COS. PC = cos. CG cos. GP. 

and multiplj'iug the first and last columns vertically, Ave find 

cos. AE cos. BG cos. CF = cos. AF cos. BE cos. CG. 

Cor. If the triangle in triqvjzdrantal^ we shall have 

tan. AE tan. BG tan. CF = 1 = cot. AF cot. BE cot. CG. 
14. We shall here give (although we forget from whom we lake it, 
and what kind of demonstration was given of it.) another such pro- 
perty of the triqiiadrantal triangle ; and the student who is versed in 
Analytical Geometry, will recognise in it the trigonometrical demon- 
stration of a remarkable property of a point referred to rectangular 
coordinates. 

Let D, E, be two points on the sphere, and 
ABC a triquadrantal triangle. Then we have 
this property, viz. 

cos. DE = COS. DA cos. AE + cos. DB -{- cos. BE 
-|- cos. DC COS. CE. 

For COS. DE = cos. CD cos. CE -\- 
sin. CD sin. CE cos. DCE .... (a) ; 
and, by right angled triangles, 

sin. CD = — — - and sin. CE = '——- . . (b^. the angle DCE 

COS. AF cos. AL ^ ° 

is measured by FL = AF ± AL ; and hence {a) becomes 

COS. DE = COS. CD cos. CE -f- ^°^' f^ . ^°^' f f 5 cos. AF cos. AL-f 
' COS. AF COS. AL < 

sin. AF sin. AL^ 
= cos. CD COS. CE -f COS. AD cos. AE qp 

COS. AD cos. AE sin. AF sin. AL 

COS. AF COS. AL .... (, > 

But, by right angled triangles, 

' Tliis remarkable proposition appears to have been discovered bv Cnrnoi, and was 
first published by him in 1S03, and afterwards in 1806, in the Geometry of Position, 
and the Esscy upon Transversals. It was subsequently and independently dis- 
covered by an eminent mathematician in this country, Mr. Whitley, who infeiTed it 
from the coiTesponding plane one, in the. Ladies' Diary, 1817. the demonstiation 
above given is taken entirely ft-orn Carnot, and it is a beautiful model for the method 
of conducting such inquiries. More ample infonnation on these subjects may be had 
in the Kepositoiy, nb supra. 




POLAR TRIANGLES. 185 

COS. AD = cos: AF cos. FD, and cos, AE = cos, AL cos. JJK ; 
also, sin. AF = cos. BF, and ^ sin, AL = cos. BL, 
hence the last term of (c) reduces to 

cos, FD cos. BF cos, LE cos. BL, 
and by right-angled triangles, the first pair of these factors is equal to 
cos, BD, and the second pair to cos, BE, and thus is the proposed 
theorem established.* 

Cor. L When D and E coincide, cos. DE = 1, and we have 
C0S.2 AD -f C0S.2 BD + cos.^ CD = 1, 
sin.2 DH + sin.2 DG+ sin.^ DF = 1. 
Cor. 2, By (9) we have 

sin, AF sin. BH sin. CG = sin. BF sin. AG sin. CH 
= COS. AF COS. BH cos. CG or, by division, 
tan. AF tan. BH tan. CG = 1 = tan. AL tan. B 1 tan. CK. 

Cor. 3. When DE = -^ we have, cos. DA cos. AE -f cos. DB cos. BE 

-{- COS. DC COS. CE = 0; (vide Young's Anal. Geom., p. 228, art. 182— 
just published by Carey, Lea, & Co. Philadelphia.) 

15. The following propositions, dependent upon what has been done 
here, or else upon similar methods, are left as exercises for the student. 

(a.) Let a transversal great circle cut any spherical polygon ; dividing 
each side into two segments; the product of the sines of the one set of 
alternate segments will be equal to the product of those of the other set. 

(b.) If a great circle bisect the angle of a triangle, (either interior or 
exterior,) the sines of the segments of the base have the same ratio as 
the sines of the sides. 

(c.) The three bisectors pass through the same point on the sphere. 

(d.) Perpendiculars to the middles of the sides pass through the same 
point, (centre of circumscribing circle.) 

(e.) Perpendiculars from the angles of the triangle to the opposite 
sides pass through the same point. 

(/.) Great circles joining the middles of the sides to the opposite an- 
gles mtersect in the same point. 

{g.) Great circles joining the points of contact of the inscribed circles 
with the sides, and the opposite angles pass through the same point. 

(A.) Great circles passing through the points of contact of the circle 
which touches a triangle exteriorly, and the opposite angles, pass through 
the same point. 

(i.) Great circles bisecting the interior angles of a spherical triangle 
meet the opposite sides in three points, which are situated in one great 
circle of the sphere, 

{k.) Show under what conditions the propositions (12) and (13) admit 
of conversion. 

(l.) Perpendiculars from the angles of a triangle upon the opposite 
sides intersect in three points, and the triangle formed by joining these 
points has its angles bisected by the said perpendiculars. 

It would have been easy to extend and to vary these subjects almost 
without limit. As the method of Transversals is the most powerful 
one yet known for the investigation of spherical determinate theorems, 
(seeming to make up for the deficiency of parallels and similar triangles, 
the great organon in plane researches,) we thought it better to dAvell 

It may be proper to mention here, that since the above demonstration was written 
I have remarked the same property in Dr. Luby's Trigonometry, p. 61-2 ; but I must 
have first met with it elsewhere, as I well recollect that it was" unaccompanied with 
any proof. Dr. Luby's demonstration is a good deal similar to mine. 
11* 



126 



SPHERICAL GEOMETRY. 



upon this sufficiently to give the student a real insight into the character 
of its processes and to furnish him with a few suitable exercises for his 
own improvement in such investigations,* 




CHAPTER 11. 

ASSOCIATED TRIANGLES. 

1. Let ABC be a spherical triangle, having its 
sides produced to meet again in A' B' C, re- 
spectively opposite to the angles A, B, C. Four 
triangles are thus formed which have a necessary 
relation to one another. These we propose to call c" 
the associated system of triangles, or simply the asso- 
ciated triangles. 

That which was first drawn (ABC), and which 
serves as a basis of the rest, we call the funda- 
mental triangle of the associated system, or simply 
the funduviental triangle. ■* 

The others, two sides of each being supplements of two sides of the 
fundamental, and two angles of two angles, we call the supplemental 
trioMgles of the associated system, or simply the supplemental triangles.i 
Moreover, when we wish to specify any one of the supplemental 
triangles, we shall do ii by reference to the side in it which is common 
to a side, or an angle which is equal to an angle of the fundamental 
triangle. Thus, to designate the triangles BA'C, CB'A, AC'B, we say 
the supplemental triangle taken with respect to A', (or a, as the case 
mav be,) with respect to B', or with respect to C. 

As a uniformity of notation is essential in inquiries like these, related 
to classes of similar objects, we shall attempt to conform to the estab- 
lished notation as a basis. Thus, abc are the sides of ABC, 
a b, c, - - - BA'C 
a,, he,,- - - - CB'A 
«//./*/// c - - - AC'B. 

In which the number of subscribed accents points out the particular 
triangle designated, considering them to be ranged round the funda- 
mental one in the order of the letters A', B', C. 
Again, for the angles, we have the angles 

A B C of the fundamental triangle ABC 

A B, C, those of BA'C 

A,, B C,, those of CB'A 

A,yy By,, C thosc of AC'B. The values of these are 

• A number of important properties of splierical triangles, demonstrated geometrically, 
by Professor Lowry. may be seen in the first vol. of the old series of the Mathematical 
Repository, and some others in Howard's Spherical Geometry, 1798. The subject, how- 
ever, is still open to indefinite rescarcli. and offers ample reward to those vsrhose taste 
may lead tliem to cultivate it. See also note A. 

t The term " supplemental" has been used by English Mathematicians to designate 
that triangle which is now universally denominated the "polar triangle." The word 
has ceased to be used in that sense for some years, and as it is so peculiarly adapted to 
express the triangles which are formed by producing the sides of the fundamental, we 
have not hesitated to adopt it. We give this notice, however, of the change in its ap- 
propriation, lest some confusion should arise in the mind of the young mathematician 
when he sees in Trigonometrical works, of the last age, a use difTerent from our own of 
the word supplemental. 

It may be remarked that the choice of the word for that purpose was not happy ; for 
though it was so far a defining property as to give the species of the triangle, it did not 
eive its position ; an element quite as important, in many investigations, as the species 
itself, and, indeed, that upon which several of its valuable properties depend. 



ASSOCIATED TRIANGLES. 



127 



b, =Tr — b) c, = n — c B, —It — B;C/ = it — C 

a„ =in — a; c,, =TT — c A,, = n — A; C,, =7r— C 

a,y, = n — a\ b,„ = -K — b A,,, = n — A ; B,„ = -r — B 

Again, for the radii of the circles inscribed in the four triangles, 
taken in the aforesaid order, we write r, r,, r,„ r,,/. whilst, for the 
ladii of the circumscribed circles, we put R, R„ R^^, R^^,, respectively. 
The unaccented letters referring to the circles of the fundamental tri- 
angles. 

These triangles possess many beautiful properties when considered 
m their mutual association, which render them worthy of greater atten- 
tion than has yet been bestowed upon them. Indeed, till very recently 
their existence has scarcely been alluded to by writers on spherical 
subjects, and even to the present day, not more than three of their pro- 
perties have, we believe, been published. 

2. Let O be the centre of the circle in- 
scribed in the fundamental triangle, and G, 
H, K, its points of contact with the sides, join 
AO, BO, CO, and draw the radii to the points 
of contact. Then the tangents from A to the 
circle are equal ; that is, AK = AH ; in like 
manner BK == BG, and CG = CH. 
rAK = AH = a 
Put ^ BK = BG = /5 
LCG = CH = y 



From which 



Whence 



■ Lc = a -f y. 




+ /? + 



^ = 



y = 



a-\-b-\- c 

2 
— a-\-b-\- c 

2 
a — b -\- c 

3 

a-\-b — c 



(1). 



Again, in the right-angled triangle BOG, we have 
tan. OG = sin. BG tan. OBG, that is, tan. r = sin. tan. § B ; or by (1) 
just given and (3), upon page 49, applied to B, we have 



tan. 


r 


" 


sin. 5 — 


* sm 


— a sm. 
s sin. s 


5 — 

-b 


_V 






"^sin. 


5 sin. 


s — 


a sin. 5 - 


-b 


sin. 5 


— c 



. . (2). 

sm. s 

Again, in the supplemental triangle BA'C, denoting the quantities 
BB', CH', and AK', by /?/, y^, a^ we shall have 

«. = /?/ + y. 
*/ = «/ + y, 

and hence, as before, s, —a,^(ij-{- y, — 

a,-\-b,-\-c, a -\- TT — b-\-7r — ^ — «-f*-j-c 

3— = 2" "=" 2 



= T 5 — «, 



128 



SPHERICAL GEOMETRY. 





-a,^b, 
2 


a — ir- 


— a-x-tr — J -f rr 


— c 


a-\-b^c- 




2 

s,—b,=P, 


b + c 


- 2 


«/ 


-b-{-n—c a-\-b 


V r 




2 

-\-b, — c^ 


a-{-iT 


2 - 2 




a^ 


S/ — C/ = y/ 

— b — TT — c a — 


= 5 — 6. 



= s> 



Also B/ = TT — B, and tan. J B, = cot. I B. Hence, in the right, 
angled triangle BO'G', we have tan. O'G'^ sin. BG tan. O'BG^, that 

sin. ssin. s — b \\ 



is, tan. r, = sin. /?, tan. ^B, = sin. s- 



sm. s — a sin. s 



Jsm. 



5 sin. s — a sin. 5 — b sin. s 



sin. (s — a). 

In exactly the same way we find the other associated inscribed radii, 
aiid the whole tabulated gives 

! 5sin. ssin. (s — a) sin. (s — J)sin. (s — c)?' 

tan.r= J < ^^ ^ ^ ^ -^ 

sin. 5 

j 5sin.ssin.(s — a) sin. {s — &)sin.(s — c)l 
tan. ?"/ — (J t 



tan. r,,= J 



tan. r. 



sin. (s — a) 
sin. s sin. (5 — a) sin . (s — S) sin. (s — c) ? 
sin.(s — b) 

I 5 sin. s sin. (s — a) sin. (s — b) sin. (s — c)l 
sin. (s — c) 



(3). 



These formulse were first given by Professor Lowry (1829), Leybourn's 
Repository, vol. v. p. 3. Multiply these together, then we obtain 
tan. r tan. r, tan. r,j tan. r,,, = sin. s sin. (s — a) sin. (s — b) sin. (s — c) . (4). 

Divide (4) by the squares of each of the equations in art. (3), the first 
side by the first side, and the second by the second : then 

sin.2 s = cot. r tan. r, tan. r,, tan. r.^^'] 
sin.2 (s — a) = tan. r cot. r, tan. r,, tan. r,,, I ,.. , . , ^p^„^t„blp 

sin.2 (s — ft) = tan. r tan. r, cot. r,, tan. 7-,,, p • • ^^^ ^^^^h remarKaDle 
sin.- (s — c) = tan. r tan. r^ tan. r^^ cot. r^^^ J 
formulae are due to Mr. Lowry (1819), vide Repository, 11b. sup. 

Again, by multiplication of the terms in (3), we have 
tan. r tan. r, -f tan. r,, tan. r,,, = sin. (5 — b) sin. (s — c) + sin. s sin. (s — a) 

. (i-\-{b — c) . a — {b — c) , . b4-c-\-a . {b4-c) — a 
— sm. — '^ ' sm. f- sm. ' ^ ' — sm.- 



c b-\^c 



2 



sin.2 -— 1= sin. b sin, c. 



Taking also each of the other corresponding combinations, we obtain 
in all the three following equations, 



ASSOCIATED TRIANGLES. 



129 



tan. r tan. ?v 4" tan. r,, tan. r,,, = sin. h sin; c i 
tan. r tan. r^, -\- tan. r^ tan. r„, = sin. a sin. c ^ . (6). 
tan. r tan. r^^, -f~ tan. r, tan. ?V// =: sin. a sin. 6 J 
Or, by addition, we have at once the following theorem. 
'tan. r tan. r^ -{- tan. r tan. r^, -[" tan. r tan. r^^, -{- tan. ?v tan. r,, -f- 
tan. r, tan.r^,/ -|~ tan. r^, tan. r^,, 
= sin. a sin. h -[- sin. a sin. c -|- sin. & sin. c . . . (7). 
That is, in words, the sum of the binary products of the tangents of the 
four inscribed radii are equal to the sum of the binary products of the sines 
of the sides. 

We may notice one beautiful theorem more, which is due to Mr. 
Lowry, ubi supra. It is tan. r^tan. 7-,/+ tan. r, tan. r^/^-j- tan. r^, tan. r,,,=^ 
sin. s ^ sin. (s — a)-\- sin. (5 — b) -\- sin. (s — c) ^ . (8) . 
For the several consequences of these theorems, and a continuation 
of the inquiry, we must refer to the number xxiv. of Leybourn's Re- 
pository, now in the press, where expressions for the various trigono- 
metrical functions of the sides and angles of the triangle, will be given 
in terms of the inscribed radii. 

3. We now proceed to consider the circumscribed radii of the asso- 
ciated triangles. We shall immediately find these in terms of the 
angles, as we did those of the inscribed in terms of the sides. 

Let CI be the centre of the circumscribing circle of 
the fundamental triangle, and draw the perpendiculars 
aM, aN, aP. Then M, N, P, bisect the sides a, b, 
c, respectively, and the several triangles BGIC, CGIA, 
AGIB, are isosceles. Let the angles made by the radii 
CIB, Q,C, with the side a be denoted by a ; those with 
b, by jff, and those with c by y.* Then A = P -{- y, 

B = a-f-y,C=a + /?. 

From which we have 



a+l3 + Y = 



_ A+B+C 


2 
— A+B-fC 


2 
A — B-f-C 


~ 2 
A-l-B — C 



= S, 




= (S- 


-A) 


= (s- 


-B) 


= (S- 


-C) 




(9). 



2 

But, by right-angled triangles BMCl, we have 
cot. CIB = COS. CIBM cot. BM, or cot. R = cos. a cot. i a-, or, by (9) 
and (p. 50-1), we have at once 

cot. R = J — cos. S COS. S — A cos. S — B cos. S — C 
— COS. S 



Proceeding with respect to the triangle BA'C, in a manner analogous 
to that employed in obtaining the three last equations of (3), using the 
values of a, /?, y, just given in (9), we shall have the following tablet of 
values, 



* These quantities being merely introduced as subsidiary ones, to be replaced in all 
general formulse by their values in terms of A, B, C, we have not transgressed our gene- 
ral rule in employing them to designate two different sets of uantities in this place as 
in art, (2) which belong finally to the inquiry. 

a 



1^ 



SPHERICAL GEOMETRY. 



cot. R = J 


COS. 


S COS. S — A COS. S — B COS. 


s— C^ 


— COS. S 


cot.R/i= n1 


— COS. 


S COS. S — A COS. S — B COS 


s— c 












cot. S — A 




COS. 




s^^ 


cot. R^^— '^ 


S COS. S — A COS. S — B COS. 












COS. S — B 




COS. 






;ot. R,,y = J . 


S COS. S — Acos. S— Bcos 


s— _c 



(10); 



cos. S — C 

which, with the following beautiful theorem, analogous to Lowry's, at 
p. 128, (obtained hy multiplying these together) is due to Dr. Lardner, 
(1826). Trig. p. 153. cot. R cot. R^ cot. R^, cot. R,,, = 

— cos. S cos. S — A cos. S — B cos. S — C (11). 

Divide (11) by the squares of each of the equations in (10), and we 
have C0S.2 S = tan. R cot. R^ cot. R,^ cot. R^yA 

cos.2 S — A = cot. R tan. R^ cot. R,, cot. R,,, 1 
cos.2 S — B~= cot. R cot. R, tan. R,, cot. R,, • ' * ' ^ ^* 
cos.2 S — C r= cot: R cot. R, cot. R,, tan. R, 
These elegant theorems, which are here published for the first time, 
were discovered by my learned friend, the Rev. H. F. C. Logan, Pro- 
fessor of Mathematics in the Catholic College of Prior Park. The first 
of them is a remarkable expression for the spherical excess in terms oj 
the four circumscribed radii. The spherical excess in terms of the in- 
scribed radii may be seen in the Repository before alladed to ; and 
some theorems connected with the same function of the triangle will be 
given in a future page of this supplement. By combining (10) in the 
same way as (5) was combined to obtain (6), we shall have 
cot. R cot. R^ + cot. R„ cot. R,,, = sin. B sin. C ^ 
cot. R cot. R,, -f cot. R, cot. R„, — sin. A sin. C }>.... (13). 
cot. R cot. Ry,j -j- cot. R/ cot. R^^ = sin. A sin. B j 
Hence, by addition, we get cot. R cot. R^-j-cot. R cot. R/^-j- 
cot. R cot. R,,, + cot. R, cot. R,, -f- cot. R, cot. R,,, -f- cot. R,, cot R,,, 
= sin. A sin. B -\- sin. B sin. C -{• sin. C sin. A .... (14). 
In the Repository (xxiv) will also be found expressions for the tri- 
gonometrical functions of the elements of the triangle, in terms of R, 
R/, R^,, R,,/ ; and we may here remark that by means of a theorem to 
be given at page 133-4 of this treatise, the expressions (10, 14, incl. and all 
of the same class) may be derived, by inspection, from those given in 
terms of sides and inscribed radii. It is by means of a property of the 
polar triangle. We shall, however, before proceeding to the theory of 
polar triangles, point the student's attention to two interesting proposi- 
tions, the analytical expressions for which we have passed by without 
particular notice. We allude to the values of tan. r, and of cot, R, at 
pages 127, 129 ; and which are 

tan. r = sin. /? tan. \R = sin. \{a -\- c — &) tan. \ B 
cot. R = cos. a cot. J a = cos. | (B -|- C — A) cot. \ a 
1, From the first of these we infer that if B and a-\- c — h are con- 
stant, r will he constant ; that is to say, in any spherical triangle if the 
vertical angle (B) be constant, as also the difference betioeen the base and 
sum of the other two sides, the radius and centre of the inscribed circle will 
continue fixed. 



ASSOCIATED TRIANGLES. 



131 



(a) be constant, as also the difference between the vertical angle, and the 
sum of the other tioo, the radius and centre of the circumscribed circle will 
be fixed ; that is, the locus of the vertex loill be a circle. 

3. This last property suggests a remarkable simple method of de- 
monstrating the beautiful theorem of Lexell which is this, viz. that if 
the base and area of the spherical triangle be consta7it, the locus of the ver- 
tex will be a circle. For, referring to the tigure at page 129, let BC be 
the constant base, and ABC any one of the triangles. Produce the 
sides to meet in A', and call the angles at B and C beloio the base B' and 
C^. For the area of the triangle ABC we have the expressions 
A + B-fC — TT = constant. But A r:::: A', B = tt — B', C = tt — C; 
hence, by substitution. A' — (B' + C) + ^ = const. .-. B' -j- C — A' 
= const, and, therefore, as the base BC or a' is also constant, it follows 
from the theorem just demonstrated that the locus of the vertex A/ is 
the circle A'BC, and, consequently, the locus of A, which is the anti- 
podes of A', must be an equal circle. "We ought to remark here that 
this demonstration is the same in substance as that given by M. Lowry 
in Leybourn's Repository, vol. i.* 

Polar Triangles. 

We have already seen, (p. 45,) that if from the three angular points 
of a spherical triangle ABC we describe three great circles, they will 
form an associated system of triangles, one of which also has a re- 
markable relation to the triangle ABC : but it has not, so far as we 
know, been noticed, that if we complete the associated system, whose 
fundamental is ABC, then the two sets of associated triangles thus 
produced, will be separable into four pairs, (one of each system forming 
a pair) which will be related to one another precisely in the same way 
as the above named pair are related. Thus, if in the annexed figure, 
ABC be the fundamental triangle, ahc its polar; then four pairs of 
polars will simultaneously be produced, viz. 

^^^ abc ! ^^^ ab'c 

Now the first pair ABC, abc 
are by hypothesis polars. t 
Hence B is the pole of ca, that 
is of ac' ; A is the pole of be, 
that is of be' ; and since C is 
the pole of ab, therefore C is 
the other pole of ab. The 
points A, B, C^ are the poles of 
the sides, therefore, of the tri- 
angle abc'. Whence, also, it 
follows by the reciprocity of 
the polar system, that a, b, c' 
are the poles of ABC. 

In the same way it is shown 
that ACB' and acb' form a 
polar system, and that BA^C, 
ha'c, form another. Hence it follows that if one 
ciated system be polar to one of another associated system, then each 
of the triangles of one system is polar to one triangle of the other asso- 
ciated system. 

* In the Edinburgh Transactions the author of this supplement has investigated 
Lexell's theorem by a novel analysis,— fAe geometry of spherical coordinates, which 
determines in a direct manner the equation of the locus, and then shows its identity 
with the previously found equation of a circle. 

t When we speak of parallel lines without specifying which is taken as the line of 




triangle of an asso- 



i^ 



SPHERICAL GEOMETRY. 



This is not the only cnrious property of the figure before us : and we 
shall put down a small selection from those we are in possession of, 
not doubting that on many accounts they will be interesting to those 
geometers who indulge in trigonometrical speculations. An addition to 
these will appear in the 25th number of the Mathematical Repository. 

Draw the great circle aA. (next figure), and produce it to m<>et BC, 
be in G and H. Then, because a is the pole of BC, and that A is the 
pole of be, the arcs aG, AH, are quadrants. Hence, aQ-\- AH = oH. 
-\- AG = IT, and the angles at G and H are right angles. / 

Let R, R', be the intersections of BC, he ; then, because xUe angles at 
G and H are right, R, R', are the poles of a A. 

Let AB meet be in K, BC meet ab in 
L, AC meet be in M, and ae meet Be in N. 

ThenZ>H-^Hc = GAC^GAB ) 



&H-I-GAB 



He + GAC. 



For JM = Z»H+ HM = 6H + HAM 
= ^H -f- GAC. In like manner, cK — 
cH + HK = cH -[- HAK = cH + GAB 



Also hM = 



= cK, since b and c are 




the poles AM and CK. 

Hence the propositions as stated are 
true. 

These are due to Professor Tiovny, and 
were given by him, in 1800, in Leybourn's 
Repos. old series. No. 1, p. 44, 5. They 
have not, that we are av^are, been no- 
ticed in any other place. 

Returning now to our original figure, let the three arcs, A«, Bi, Cc, 
be drawn ; these will pass through the same point O, because as has 
been just shown, they are perpendicular to the three sides BC, AC, AB, 
(vide p. 125). Also, because «A is perpendicular to be, it will pass 
through the opposite pole a'. In like manner it will pass through hf. 
Or the points ahJda' h! are in the same great circle, whose poles are 
R, and R', the intersections of BC, be. 

In like manner 6B0i'B' are in one great circle, whose poles are S, S', 
the intersections of AC, ae\ cCOc' C' are in one great circle, whose 
poles are Gl, Gl', the intersections of AB, o]). Again, because R, R', are 
the poles of aa' the arcs RO, OR', are quadrants. In like manner SO, 
OS', and Q.0, 00.', are respectively quadrants ; and as the quadrants 
are drawn from the poles of aoJ , hb', ee', respectively, R, O, R' ; S, O, S', 
and Q,, O, Q.', are respectively in the same great circles. 

Also since 0R= OR' = OS = OS' = Oa = Oa' = -^, the points 
R, R', S, S', A, Q,', are in the same great cirele, and O is its pole. 

The complexity of the figure which would represent from a further 
detail of these interesting researches, compels us to leave them for the 

reference, and knowing that the second is related to the first in the same way that the 
ifirst is to the second, we simply denominate them parallels. The same practice also 
holds in speaking of two mutaally supplemental angles. But when we previously fix 
upon one line or one angle, we say the parallel, or the supplemental in the singular 
number to express the other line or angle. Just so in respect to the two triangles 
which constitute the polar system, when we speak of both without assigning the refe- 
ree, we call them polars, as a common epithet ; but when we have fixed upon one 
already, as that to which the other is referred, we call it the primary and the other the 
polar triangle. 



ASSOCIATED TRIANGLES. 133 

present to the industry and ingenuity of the student. They are exceed- 
ingly easy and furnish an excellent exercise in spherical investigations: 
and we therefore hope he will give it a proper degree of consideration. 
"We proceed to view the subject algebraically. 

Let abc, ABC, be the sides and angles respectively of a spherical 
triangle ; Rr, the radii of the circumscribing and the inscribed circles ; 
s, S, the semi sums of the sides and angles respectively ; also let a' b' c' ; 
A/ B' C ; R^, r' ; S', s', be the same things in the triangle which is 
polar to aZ>c. Put, as at page 116, 

723 = sin. s sin. s — a sin, s — b sin. s'-^—c 



sin. 5 sin. s' — a' sin. s' — b' sin, s' 



W=: — cos. S cos. S — A cos. S— B cos. S— C 
N'2 =— cos. S' cos. S' — A' cos. S'— B' cos: S' — C. 
Then we propose to prove that n^ =N'"2, and n'^ — W. 
For s — l( a-\-b-\-c) 

s — a = l\ — a -\-b-{- c) 
s — i = i( a — b -\-c) 
s — c = §( a-\~b — c); and, in the polar trls 

igle, we have immediately S' = -^r i {a -j- b -\- c) = ~ — s, 



A.^=^-i{-^ + b + c) = - 



S' — B' = — — !(« -b-{-c) =— — s — b, 



o 

Hence, — cos. S' — — cos. ( — s) = sin, s, 

2) 




a) = sin. 5 — «, 



b = sin. s — b, 



cos. S' — C — cos. (— 5 — c) = sin, 5 — c. 

Whence,. by multiplication of the extreme vertical columns we have 
the quantities designated by N'^^ ^nd n^ ; or, N^^ _ yj2_ jj^ exactly the 
same way, we find that N^ = n' ". That is, N' = n, and N = n'* 

* These properties might have been more simply obtained, thus : by 
(18) and (21), page 117. 

n' N 

tan, h A' tan. h B' tan, h C — — : — -^, ;cot, i a cot. \ b cot, | c = — — -. 
sm.^S' cos.^S 

But \ (A' -{- a) — —- ,-, cot. \a — tan.-— A', and hence the left sides 

are equal. Also, cos. ^S = sin. ^s, therefore N =: %'. 

Had the mere proof of the property been our object, this would have 
been the preferable method : but the stages through which in the text 
we have passed are necessary in other inquiries, and we prefer there- 
fore to give them in that form which, it may be remarked too, is the 
original one by which the property was obtained. See note B. 
12 



134 



SPHERICAL GEOMETRY. 



Again, by Lowry's theorem, page 128, 

tan. r =: —. , tan. r' = ; ; and, therefore, by (17, p. 117), 



sin. 57 



tan. r' 



N' 



N^ 



2 COS. i A' COS. i B' cos. i C 
Also (10) p. 130, and (19) p. 117, 



I sin, I a sin. i b sin, | c 



tan. R = — 



COS. S 2sm.iasm.ibsm.ic 



tan. R'=:- 



cos.S' 



N' 



„.r, „ . , -^-r tan. r 2 sin. i a sm. J S sm. i c 

Whence, recollecting that N = w, — — = - — -. — tt — r^ — , — v^ — 

tan. r' sm. ^ (a + 4- c) 



and 



tan. R 2 sin. i a sin. J 6 sin. | c tan. r 



tan. R' sin. ^{a-\-b-\-c) tan.r' 

But we have a more useful result, as follows : 

„ 2sin. J ffisin.|Z»sin. I c N' 

tan. R tan. r' — 



tan. R 
tan. R' 



h 



n 2 sin, ^ a sin. § h sin, J c 

or tan. R = cot, r', which is fulfilled by the relations 

R -{- r' ='-5-, and R -|- ^' = -77-- The former of these relations holds 

when the circles are referred to their nearest poles, and the latter when 
to their farther poles. In like manner, we also have, in the corres- 
ponding cases, R' -|- r = -^ and 'Rf-\-r=:-— 

From these we get R + '?''= R-' + ^ 
R— R'= r —r* 
R— r =W—r'. 
Thus the remarkable relation between the polar triangles is continued 
even amongst the radii of the inscribed and circumscribing circles, viz. 
that the inscribed, radius of one is the complement of the circumscribing 
■radius of the other ; or, taking diameters, the supplement of the other. 

We have seen that in the two associated systems, having one pair of 
mutually polar fundamental triangles, the two systems, are triangle for 
triangle"mutually polar ; and hence collecting the whole result into one 
table, we have 



R +r 


= 


2 


R +r' 


= 


n 

"2" 


R^+r' 


= 


X 

"2" 


R/ -\-r', 


=r 


~2 


R',/ + ^// 


= 


'2 


R., + '>•'.. 


= 


IT 
'2 


R'.,+ r,.. 


= 


TT 
"2 


R.. + r'.. 


= 





R' +r = 

R +r' = 

RO+r, = 

R/ +^/ = 

R'// + ^// = 

R,/ + r',, = 

R'//y+^/// = 



h 

2 

3n 

2 

Jf 
2 

_?! 
2 

2 

3l 
2 

3rr 



Hhr'.. = ^ 



ASSOCIATED TRIANGLES. 1^ 

gether, we have 

r-\-r,-\-r,, + r,,,+ r' + r', + 7-,, + r',,, + R -^ R, + R,,+ R^,,-]- 

R' + R',+ R^, + R^,, = 47r (or 12 tt). 
TAai w, ^Ae si^w of all the sixteen radii of the primary and polar asso- 
ciated system is eqtial to the surface of the hemisphere if the first system of 
values is taken, or to three times that sum if the second system be taken. 

The consideration of the latter system of values I owe to the Rev. 
Professor Logan ; the former with its results is my own independent 
discovery, and was the origin of my researches on the subject of polar 
triangles, and of the associated triangles too. 

Another property also comes immediately from this : viz. that the pro- 
duct of the tarigents of the sixteen radii is equal to unity. Which is seen 
at once by tan. r. tan. R' = 1, &c. : or it may again be put 

tan. r tan, r^ tan. r,, tan. r^,, tan. r' tan. r', tan. r'^^ tan. r,,, = 
cot. R cot. R, cot. R,y cot. R^,,. cot. R' cot. R'^ cot. R',, cot. R'^//; 
or still differently, 

tan. r tan. r^ tan. r^, tan, r^^, cot. r' cot. r"^ cot. r',y cot. r^,^, 

cot. R cot. R, cot. R,, cot. R,,, ~ tan. R' tan. R', tan. R%, tan. R',,,' 
Another neat relation may be put down here ; we have already seen 

tan. R 2 sin. | a sin. i b sin. | c ■, . • . . • .i, 

that —, = : — —, — r-,—. — ^— ; and by mterchangmg the pri- 

tan.R' sm. i {a -{- b -{- c) ' ^ 

mary and secondary polar triangles, still retaining the accent upon the 
same letters to distinguish them as the sides of the same triangle as 

, - , tan. R' 2 sin. J a' sin. i b' sin. | c' 

before, we have ^=— = : — ; — - — . 

tan. R sm. J {a/ -\-h' -{- c') 

Dividing these, we have 

tan, 2R ^ sin. \ a sin, \ b sin, \ c sm. \ {a'Ar b' -f-Q 

tan. "^R' sin. |<z'sin. i^'sin. Jc' ' ^\v\..\{a-\-h-^6)' 

But \a = — i A, &c. because the triangles are polar ; 

tan, ^R _ sin. \ a sin. \ b sin. \c — cos. S 

' ' tan. 2R' — COS. * A cos. \ B cos. \ C ' sin. S~ * 

sin. \ a sin. \ b sin. \c — cos. S T»/r , • i • 

= : — ^ • r-7 r^ tth- Multiplymg the 

sm. S COS. I A cos. * B cos. | G r j & 

same equations we get 4 cos. ^A cos. \ B cos. |C sin. \a sin. \b sin. \ c 
■= — cos. S sin. s ; or in every spherical triangle we have 
2 COS. h A cos. I B cos. | C _ sin. s 

COS. S ~ 2 sin. J a sin. k b sin. ^ c ' 

which, by the bye, is also an immediate consequence of the relations 
(17), (19), at page 117. Innumerable other interesting results may be 
obtained with equal facility, by means of the property of polar radii 
given above ; but the limits of a work, like the present, prevent our 
enlarging upon them here. "We may, however, refer for some of them 
to the Mathematical Repository, No. xxiv. 



186 ^ SPHERICAL GEOMETRY* _^. 

CHAPTER III. 

Some additional inquiries respecting the spherical EXCEssr. 

We shall now devote a short chapter to some miscellaneous inquiries' 
Respecting the Spherical Excess, in continuation of what has been 
already done in Chapter ii. Part iv. All the usual formulae for the 
spherical excess have there been amply discussed ; but there are still 
certain other combinations of data which ha,ve not yet been considered : 
these are, 1st, Two angles and the interjacent side; 2d, Two angles and 
a side opposite to one of them : and lastly. Two sides and an angle 
opposite to one of them. Expressions for the spherical excess in each 
of these cases may be readily deduced. In the last two, however, the 
formulae which I have obtained are neither sufficiently symmetrical nor 
sufficiently simple to render them deserving of much notice, either for 
ianalytical beauty or for practical convenience ; they involve, how;every 
but one radical. The formula for the first of the above ca:ses I inves- 
tigate as follows. 

To determine the Spherical Excess when two angles and the interjacent- 
side are given. 

„ ^ E A + B + C— 180° 
Here we have— == — ■ !— , 

^ , . - E A+B . C , . A-f-B C 

and therefore cos. -r- = cos. — ^- — sm. -7r-+ sm. — — cos. -— ,. 

E A+B C A+B . C 

sm. — = cos. — ^- cos. sm. sm. -^. 

C C 

But sin.2 — - and cos.^-^ may take either of the following forms : 
2 Z 

. „ C I — COS. C 1 + cos. Acos. B — sin. A sin. . Bcos.c 
sm.** -TT — — = — ■ — 




2 

.A , B „ A . A 

=± C0S.2-- C0S.2 — 2 cos. -^sm.-— - cos. 

Z Z 'Z z 

. A . ^ B , A — B . ^c . 

sm.2— sm.- — = C0S.2 - — - — sm. ^ -^ + cos 
Z Z 'Z z 

„C 1 + COS. C 1 — COS. A cos. B + sin. A sin. B cos. c 

•^^^-V^ — 2 2 • 

A . B „ A . A B . B , . oA „ B 

«= cos.2 - sm.2 - -2 cos. -sm. - cos- - sm. ^ cos. c + sm.'^ - cos,^ — 
z z z z z z z z 

. , A — B . ^ c , . oA + B „c 

== sin.2 — - — sin.2-— + sin.2 — cos.^-— . 

^ z z z 

Thus we shall have a choice of three forms, to suit the specific pur- 
pose we have in view. The last is the preferable on the ground of* 
algebraical symmetry. Substituting these, we have 

E A + B, A — B . , <; ■ 2 A+B „ c ?J 

COS. —• = COS. — i — J C0S.2 — - — sm.^'— ■ + cos.^ — ^ — cos.^ - \ 

, . A + B,.A — B.,c , .,A + B ,ci 

. E A+B, . ^ A — B . , c . . ^ A + B_ ., c I 

sin.-g =cos. — ^_ Jsm.2— ^-sin.2 - + sm.^ _— — cos.^ ^J 



SPHERICAL EXCESS. 137 

, . A + B ^A — B . ^c , „ A+B .0 >§ 

+ sin. -^- \ cos .2 — 2 sm.2 ^-+ cos.^ — ^ cos.^ - $« 

To determine the excess^ when the three sides are given. 

This case has been already discussed, but the following investiga- 
tion may not be unacceptable. By (46, 4,) 

^ . A + B+C — TT A + B — C+^ 
E 2sm. -in_f COS. -:n__ 



tan. — = '- 
4 

sin. -I - 


2 COS. 

- sin, 


A+: 


B + C 

4 

4> 


7! 

sin. 

cos. 
sii: 
cos 


cos. 

A + 
2 

A + 
2 


A 
B 

"b 


+ B 

— cos. 
-{- sin. 


4 

C 

2 
C 

2 


+^ 
• or by (ar 


t ftfi'N 


COS. 


A+B 

2 
a—b 

2 
a-^b 
2 ' 

cot. 


■f-cos. 
-cos, 
cos. 


c 

T 

C01 

c 

~2 


4= 






s — 

'• 2 


a 

-s: 

COS 


. s — 


-6 




cos. 


^^- 2 

s — 

'■-2 


c 


.(^> 


But, 


1 
= J- 


sin. s sin. 


s — c 


._ 





sin. s — a sin. 5 — b 



s s . s — c s 

sin. -— - cos. -rr- Sm. -;; COS. 



(*). 



_ 2 2 2 2 

s — a s — a . s — b . s — c 

sm. -^- COS. — - Sin. -^ sm. -^ 

Inserting {b) in (a), we have, after slight reductions, 

E I 5 s — a s — b s — c,.,., 

tan. -r- == /si tan. -^ tan. — ~ — tan. — -- — tan. — - — which is the 

remarkable formula of Lhuillier* Applying this to the polar tri- 
angles, some interesting results may be obtained as follows : 

Denoting by S,, S^„ S^^^, the semi-sums of the sides of the supple- 
mentary triangles ; by <^, b,^ and c^, the sides of BA'C ; a,,^ b, b^„ the 
sides of AB'C ; and by a„„ b,,,, c, the sides of AC'B. Then (see p. 

^97^ ^ a-{.„ — b-^n—c _ a—b— c 

127) s,_ - _ + ._ 

— a-\-b-{-c 



a — b — c a-{-b- 

b, z= \-1( IT b := 



* The excess has also been obtained, by means of the modern analysis, by Euler in 
the Memoirs of the Royal Academy at Berlin, vol. ix. p. 256, and by Tedenat, in Ger- 

gonne's Annals of JIathematics, vol. vi. p. 48. 

12* R 



139 



SPHERICAL GEOMETRY. 



a — b — c a — b-\-c . 

s, — c, = — — \~TT — TT — c = ~ — = 5 — bi 

Hence if E„ E,„ E,,,, denote the excesses of B A'C, AB'C, BCA, we 
have by Lhuillier's theorem, 



tan. E, 

4 

that is ; 


= J tan. 


S, Sy ay Sy by Sy Cy 

—tan.—— tan. ——tan.—— 


tan.^ 
4 


= ] cot. 


s . — a s — b s — c 


E,, 
tan. — 


= J cot. 


5 s — a s — b s — c 
2 *^^- 2 '°^- 2 ^^''- 2 


E.. 
tan. — - 


= J cot. 


s s — a 5 — b s — c 
-tan. g tan. ^ cot. ^ 



s — c 
-tan. — - — 



(1). 



JK ! s 5 — ff s — 

tan.— = J tan.— tan. -— - tan.— ^ 

in which Lhuillier's theorem is applied to each of the triangles in suc- 
cession. If we multiply these together, we find 

E^ E. Eyy Kyyy S 5" «^ ^ — ^ ^ ^ " ^^ /'0\ 

tan. --r-tan,---tan.-— tan-—^ =cot.— -tan. — ?r— tan. — -—tan. ^ . (2). 
44 4 4 2 2 2 2' 

Again, the angles of the triangle BA,C are 

A, = A, By = TT— B, and a =^—C. 
E^ ;r+A — B — C 

4 
E. 



Hence similarly 



Also HrA-B-C 
4 



4 

TT— A+B 


— C 


4 

TT — A — B 


+ C 


4 

— TT+A + B 


+ c 


4 

:r — A + B 


H-c 



. (3). 



whence 



TT — A+B + C E, 

tan. -^ ■ = cot. -r- 

4 4 



J tan. 



tan. — ;; — cot. — :^ — cot.- — - — ; and by similar pro- 



2 2 . 2 2 

cesses with the other triangles, we get the following table ; 



A-fB + C-frr I 5 s—a s—b s — c 

- J tan. ^tan. — - — cot. — — cot. -^ 







4 


tan. 


A — 


B + C+. 




4 




A + 


B — C-fTT 






4 




A+ 


B-f C-TT 






4 



=i 



s s — a s — b s — c 

tan. 2"cot. ~^~ ^^^- ~2~^^^'~~2~ 



s s — a s — b ^ s — c 

= ^ tan. —cot. -^ cot.— ^ tan. -^ 



I s s — a s — b s — c 

:j tan. -tan. -^- tan.-^— tan. -— ^ 



K4). 



SPHEEICAL EXCESS. 



The last of which is the common form of the area of a triangle given 
by Lhuillier, applied to the fundamental triangle. 
Multiply all these together, and we shall have, 



^ a-\-b-JfC A + B + C 

tan .2 ' , ' = tan. — ■ ^ — 

4 

A— B+C + 7 



tan. 



A+B— C+ 



tan. 



tan. 



A-f B+C+^ 
-4 



(5). 



Also, giving to the terms of Lhuillier's theorem, their unabbreviated 
values, we shall see a striking analogy in their general form between 

that and the one just obtained. For in tan.^ — — — ^~ = 



a4- b-\- c a-\-b 

tan. — '—-- — tan. — ^- 
4 4 



c a 

- tan. — 



i+c 



tan. 



■i±^i (6). 



we see the only difference, as to general form, is, that tt enters into all 
the angular functions, and not into those of the sides. Again, since 
the three last factors in the right hand member of equation (5) are 

E... E,. ^ E, J v . . ^ . E 

cot. —7- , cot. — , and cot. — ; and the remammg lactoi is tan. — , 

we have (5) converted into 
a-\-b -\- c 



tan. 



E 

tan. -— cot. 
4 



E/ ^ E//. E,/^ 

—r cot. ~ cot. — — 

4 4 4 



. . (7). 

By the principle of the symmetry of the triangles, and of their ex- 
pressions, we at once infer from (7) that 

E , E, ^ E,, E,,, „ 5, 

cot. —r- tan. -7- cot. -j-- cot. -; — = tan.^ -^. 
4 4 4 4 2 



But 



a-j-TT — b-\- -rr- — C 

2 = " 



— a -{-b-\-c 



tan. -^ = cot. \ ^'— 

2 4 



cot. 



Applying the same principle of reduction to the other supplemental 
triangles, and collecting the results, we have 



tan. 


2 


s 


— a 




2 


COt.2 - 


— b 



cot.' 



2 
c 



cot. 


^eo.. 


E,, 
4 


cot. 


E.. 
4 


tan. 


E. 

-r- cot. 

4 


E.. 
4 


cot. 


E.. 
4 


cot. 


E, 

-7- tan. 
4 


E,, 
4 


cot. 


E., 
4 



2 



tan, -r- 
4 

E 

: cot. —r- 

4 



= cot, I- cot. 4' cot. 5^ tan. ^ 
4 4 4^4 



(8). 



Let us resume equations (3), and multiply by (4) then we have 
A + B + C — rrzzzE 1 
A_B-C+,r = E, I ■ ,Q. 

— A-f-B — C + 7r = E,, r '^^^• 

-A-B+C + 7r-E,.J 

add them, then E + E, + E,, + E,,, = 2 r (10). 

Add the first of these to each of the others successively, then 



140 



SPHERICAL GEOMETRY. 



tA = 



E + E, 



E+E, 



-— ,andC = 



E+E, 



(11). 



„ A + B + C 3E + E, + E,,+ E, 



•, or, inserting the value of 



E, + E,, + E,,, from (10), it becomes S 



^ 2 



and, in like man- 



r, 



ner, we have 



A 



B = 



E., + E, 



S — C 



4 
E H- E, — E,, 


H-E,,, 


4 

E + E, + E, 


-E,,, 



(12). 



.*. — cos. S. = sin 


E 

2 

E. 

2 

E,, 

2 

E., 

2 


cos. S — A = sin. 


cos. S — B = sin. 


cos. S — C = sin. 



(13). 



Inserting these values in the usual formula for finding a side, we get 



cot.2 J a = 



. E . E, 
sm.-sm.- 



cot. 2 \ h 



and cot. ^ i c = 



2 ^-(14). 



;m. 



sm. 



sm. 



E. 



sin. 



cot 
2 



E 

2 2 

2 1 a cot. 2 i Z) cot. 

H _ 2 

E 



he 



sm 



sm. 



sm. 



E, + E,, 4- E,, 



. . (15). 



The sides and angles of the triangle are thus found, (the angles in 11,) 
in terms of the areas of the four triangles : and the equation of condition 
also which subsists among these four triangles is assigned in (10). 

By (12) the values of the factors in N are found, and by (8) there is 
another trigonometrical function of the factors of n assigned. From 
this, those factors themselves may be assigned, but the process is 
troublesome and the result inelegant. We have obtained a simpler 
form, but even then neither the form nor the method is well suited to 
this place; The values of the inscribed and circumscribed radii in 
terms of the excesses will be discussed in the Repository, and we shall 
conclude this section with assigning the connexion between the polar 
systems of associated triangles, in respect to the areas. 

The sides of the primary fundamental triangle being a, b, c, we have 



SPHERICAL EXCESS. 



141 



* 2S' = 3 ^ — (a + 6 + c), and hence — 

E' _ 2rr— (a -f-&-[-c) _ ir_ __« + Hl£ 

~~ ~2 4 



and tan. — - = cot, 
4 



cot. 



Again, in the supplemental 



4 ~ 4 

a-\-b-\-c 
• 4 =- 

polar triangles, (that taken with respect to A for instance,) we have 
2S'',= (tt — /j)-|-(ff— TT — i)-f-(ff— TT — c=7r — a-f-^>4-c 
E' tt+Ctt — a + J + c) _— a-\-h-\-c _ {s — a ) 

7= 4 ''- 4 

a 



2 



E' s 

and tan.—- = tan. - 
4 



E' E' 

and 2S'^,^, we have the values oi~- and — - 

which are collected together below. 

E' s 

tan. -r- = cot. -r- 
4 2 

¥/, s — a 

tan. -7- = tan. — - — 
4 2 

E',, 5 — b 

tan. -7— = tan. — - — 
4 2 



In performing the same changes upon 28',/, 
- ; the whole form of 
1 



tan. 



= tan. ■ 



(16). 



"P' TT' 17' E' 

Multiply these together ; then tan. -j-tan. -j^tan. -— ^ tan. — ^ 

s s — a s — b s — c ,,-. 

= cot. — - tan. — - — tan. — - — tan. — - — . . . (16). 

/i Z ^ Z 

And, again, by comparison of (2) and (16,) we find 



E' E', E',/ E', 

tan. -r- tan.-p tan. -— tan. — ^ 

4 4 4 4 



E hi, E/y E///,,_^ 

tan. -7- tan. -f tan. -^ tan.— f-'(17). 



4 4 4 

By a comparison of the component equations of table (15) with those 
of table (8), we shall get the value of the area of any triangle in terms 
of the associated system which is polar to it. Thus, 



cot.2- 



E E/ E// E/, 

tan. -— cot.— r-cot. -— cot. — ~ 
4 4 4 4 



cot.2-^ = cot. ^ tan. ^^cot. ^' cot. ^'' 
4 4 4 4 4 

o E',, E E, E,, E,,, 

cot.2 — - — = cot. -— cot. — tan. — cot. ~-r- 
4 4 4 4 4 

cot.2-5^ = cot. -I cot. 5^ cot.?-'^ tan. 5^ 
4 4 4 4 4 



(18) 



and conversely, by interchanging the system of reference in the polar 

• The accented letters S', E', ^c. denote quantities in the polar triangles which are 
denoted in the primary by S, E, Spc. 



142 



SPHERICAL GEOMETRY. 



triangles we have cot.^ --- = tan. -^cot. ~ cot 



cot. 



E^.^ 



cot. 2 
cot 



4 

E, ^ E', E^ ^ W,, ^ E', 

— = cot. —tan. ^ cot. ^-^cot. — 

„ E,, W E' EO, E',,, , 

2 — i' = cot. -- cot. — - tan. -- — cot. — r— 
4 4 4 4 4 

« E/,/ E E , E // E /// 

cot.^^ii' = cot. -pcot.-^ cot. — ^-tan. — 7-— j 



va9) 



.(31) 



4 4 4 4 4 

By multiplying either of these sets, we should also obtain the reci- 
procal of equa. (17). By means of (10) applied to both systems of asso- 
ciated triangles, we have E + E. -[- E,, + E,,, + E'-{- E', -j- E',, + E',„ 
= 47r = surface of the sphere .... (20). 

E E' 

Taking the values of tan. — tan. j&c. from (1) and (16), we get 

„E „E' s s — a s — b s — c" 

tan.2 -— tan.2 -— - — cot. — tan. — r — tan. -— - — tan. — - — 
4 4 2 2 2 2 

„ E/ „ E' 5 s — a s' — b s — c 

tan.2-— tan.2 — - — cot. — tan.— - — tan. — - — tan. -— r — 
4 4 2 2 2 2 

„ E// oE' s s — a s — b s — c 

tan.2 — - tan.2 -_ =cot.-— tan.--— tan. — -— tan. — - — 

^ „Eyy. pE',// 5 5 — « s—b s—c 

tan.2-— tan.2 -- = cot.-^tan.— ;— tan. —- tan. —^ ^ 

From the equality of the right sides of the last equations we find 

E ^ E' E, E' E,, EO, 

tan. -7 tan. — = tan. — tan. — - = tan. — tan. — — 
4 4 4 4 4 4 

= tan.^^^tan.^^i^ . . . : (22). 
"When the geographical positions of the three angles'of a spherical 
triangle are given to determine the area, we have the following expres- 
sion, first given by the author of this Supplement, in the 12th volume 
of the Edinburgh Transactions, viz. 

1 + cos. a, COS. a,y -{- sin. tty sin. a,, cos. (/?/ — /?//)^ ^ 
-\- cos. a, cos. tty,, -f sin. a, sin. a^^, cos. (/?, — ^///) 

+ cos. a^, COS. a^.y 4" sin. a,, sin. a,,^ COS. (/?^^/ — /?//) 



E 

COS. —-= J 



2 (1 -f- COS. a, COS. a///-}- sin. a, sin. a,^, COS. /?, • — /?^/) X 
(1 -\- COS. a, COS. a,/, -j- sin. a, sin. a^^^ COS. ,5^ — (^///)X 



23 



l^ (1 ~{- COS. 0/,CQS. o,/, -j- sin, a„ sin. a^,, COS. /?/^^ — /?//) 

where a,, /?, ; a,,, /?^/ ; and a^,,, /?///, are the colatitudes and the longi- 
tudes of the vertices. 

The spherical excess may also be very readily exhibited under 
another form by a direct investigation, but which, in my paper, in the 
Mathematical Repository now publishing, is obtained by inference 
from another property. Thus, 



A + B + C — rr 



A + B+C 



cot. 



A + B + C — rr 



A + B + C 



SPHEKICAL EXCESS. 



143 






o 
o 

4- 






o 
o 

+ 

o 
o 



PQ CM 



<1 ICQ 



o|c^ 



pq P 



<\< 






-:: B 



CQ rf 



S £i 



<j p 




'^5 


^ 
1 




o 




+ 


o 1^ 


1 


(1) 

p., 


d 


« 


CD 


•S 



pq|CT 



<^P 



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144 SPHERICAL GEOMETRY. 

We are also enabled, by means of the last set of equations, combined 
with (10), to ascertain the relation that subsists among the four radii r, 
r^, ^//, ^///j a relation which I believe has never before been assigned. 

E+E,-fE,, + E,,, ., 

For we have — ■ ^ = ?r ; and hence, 

tan. ~2 _ u _ i_s4 (^ ^^) _|- s^ (^ t, t,, tt,,,) ' 

Now this may be fulfilled in two different ways, either making the 
denominator infinite, Avhilst the numerator is finite; or making the 
numerator zero, whilst the denominator is a real quantity, finite or 
infinite. It would exceed the confined limits of a work like the present 
to discuss the circumstances of this function, and I shall, therefore, 
assume (though I shall elsewhere prove it) that the only condition that 
obtains is the latter. We hence have E4 (t) — L4 {t t, t,,) — ; which 
may be written as follows : 

P Ti' E E 

(having previously divided it by tan. -— tan. ~ tan. -~ tan. o^^O* 

— cot. 

— cot. 

— cot. 

— cot. 

Insert for these cotangents their values, and reduce the expression to 
its simplest form. The work is somewhat laborious, but the result is 
comparatively simple; and hence I shall leave it as an exercise, for 
the student to perform alone. In another place I have given a difterent 
investigation of this formula, and several collateral topics are also com- 
bined with it, which will render it needless to enter into further detail 
upon this class of subjects, in the present necessarily very incomplete 
sketch. I trust, however, that enough is done to excite the interest of 
the mathematical student, whilst the extent of the subject itself will 
afford sufficient exercise for his ingenuity, and reward to his perse- 
verance. 

I am obliged to terminate these researches abruptly, on account of 
the space which they would occupy, if developed with any approach to 
completeness. I take the opportunity afforded me by reading the 
proofs, to state that my friend and neighbour, the Rev. Professor Logan, 
has also engaged in these and several collateral researches, and that 
the results to which both he and 1 may ultimately be found to have 
arrived, upon comparison of our mss., will be published in a single 
dissertation to be considered as our joint production. These researches 
will extend to every other function of parts of the spherical triangle, as 
well as those which have been in this supplement discussed ; and to a 
considerable extension of each of these. It will then be seen that 
Spherical Geometry offers one of the most ample fields of research that 

* It is left for the student to prove, from the expression for the tan. 
of the sum of two arcs, at p. 3.3, that the expression for the tan. of four 
arcs is that in the text, in which E {t) denotes the sum of the tangents 
of those arcs E {tt,), the sum of their products taken two and two, and 
so on. The same may be generalized for any number of arcs. 



E , E, 


E,, E,, 


— + cot. — cot 


f'^'-i" 


E, , E 


E,, E,,, 


—+ cot— cot. 


f''2- 


W , E 


E, E,,, 


- — h cot g- cot 


^'''■t 


E,,, E 


E, E„ 


— +cot. — cot. 


r""'- T 



SPHERICAL EXCESS. 146 

has yet been discovered ; and I hope I shall not be thought too sanguine 
in anticipating that the properties of figures, traced upon the surface 
of the sphere, will, in a very few years, become as familiar to English 
Geometers as the correlative figures in piano now are.* 

* Not only have Spherical Geometry and Spherical Trigonometry been greatly 
neglected in England, but also upon the Continent. The continental Geometers have, 
however, been tnily assiduous in the cultivation of the Geometry of three divisions, 
and have imagined and discussed almost every variety of method lor conducting their 
investigations in this branch of science : whilst, on the other hand, it will be difficult 
to point to any one British Geometer who ever added a single important theorem to our 
stock, much less devised a single original method of investigation. Of the causes of 
this humiliating fact, the present is not the place to speak. It may, however, be allow- 
ed me to mention what appears to be a barrier to our removing the discreditable 
charge. We have no work, expressly devoted to the subject, in which either the 
methods themselves are developed, or the spirit of them at all displayed. Mere illustra- 
tions, taken in a considerable degree at random from different works, in which they 
were originally very appropriately placed, when brought together without due regard 
to the principles themselves, and often without adapting the notation to any uniform 
standard — works like these, though they may be entitled treatises on the Geometry of 
Three Dimensions, can scarcely be called so without a complete pvrversion of the use 
of terms. He that renders a method of investigation intelligible, witli whatever paucity 
of mere illustration, does more for the interests of science than he who collects all the 
illustrative examples of those methods that have ever been given into one single mass. 
Such cellections are, indeed, too commonly calculated to confuse the young mind and 
to repress all the ardour it might otherwise have felt. 

Long ago, impressed with the importance of the subject, the author of this supple- 
ment formed the ambitious project of supplying this desideratum, and of furnishing a 
work in which the spirit of the methods which have been employed by the continental 
Cfeometers should be the fii'st object of his anxiety. It has been his special aim, during 
the preparation of his work, to explain the essential character of each general pi'inciple, 
and to show wherever they really differ from one another, and at the same time to 
illustrate each by a siifficient number of apposite examples, strictly adapted to the pur- 
pose for which they were employed. By proceeding thus with every method that has 
been proposed by the different continental Geometers, and by furnishing also considei-- 
able portions of original results, it is hoped that a work may be produced which will 
render the study of solid Geometry scarcely more difficult than the more recondite 
portions of plane Geometry are now, and thereby gi-eatly extei:d the cultivation of that 
omnch of science in England. 

He has been, however, led to think that a subsidiaiy elementary work on Descriptive 
Geometry would not be unacceptable to British Geometers, befoi'e the other goes to press. 
Even on this, the simplest of all the forms under which the Geometry of tliese dimen- 
sions presents itself, a merely graphic form — we have no treatise in England, nor yet a 
single chapter in any English course of Mathematics. There was indeed published in 
America, in 1821, a tliin octavo, by M. Crozet, for the use of the Military College of the 
United States ; but it would be scarcely less difficult to devise tlie methods originally 
than to acquire them from that treatise. Such a volume will therefore be sent to press 
with all convenient speed, the avant courier of the larger work. 

13 S 



NOTES. 



Note A. f. 126. 

The following pretty theorems I have received from Mr. Lowiy, 
since the first chapter on Spherical Geometry was in forms. 

" Let ABC be a spherical triangle,* D the middle of one of the sides, 
AC ; and let AB = d. Then cos. a -\- cos. c = 2 cos. | b cos. d. 

^ cos. a — cos. lb cos. d -n-r^/-. 

For : ; — ^ . = cos. BDC 

Sim. i b sm. d 

cos. C cos. i- i cos. <Z -n-r-w A T»-r»/-. 

: \ = cos. BDA = — cos. BDC 

sm. ^b sm. a 

Hence cos. a cos. |6 cos. d = — cos. c -f- cos. ^b cos, d, 

QY cos. a -\- COS. c = 2 cos. ^b cos. d. 

Cor. 1. When the triangle is inscribed in a semi-circle, the diameter 
of which is b, cos. a -[- cos. c = 2 cos.^ ^b, or cos. a -\- cos. c = I -\- 
cos. b. jj 

Cor. 2. And when a =c,we have cos. a = cos.^ -^. 

Cor. 3. Hence, in a spherical square,t the cosine of the sides is equal 
to the square of the cosine of half the diagonal. 

Cor. 4. The sine of half the area of the triangle ACB in the circle is 
a, c 

= tan. -— tan. — -. Vide form 20, Math. Repos. v. part i. p. 7. 

Cor. 5. Hence in a spherical rectangle,^ the sine of one fourth of the 
area is equal to the rectangle of the semi-tangents of the two sides, thiit 

a c 

is = tan. — - tan- -^ . 

2 2 ^ 

Cor. 6. And in the spherical square, the sine of i area = tan.^ -^. 

Cor. 7. In a spherical parallelogram,ir the sides of which are a, b, c, d, 
and diagonals h, h\ we shall have 

cos. a -j- COS. b -\- COS. c -f- cos. d = i cos. --r- cos. ---. 

These properties are very simple, but neat, and might serve as ex- 
ercises in an elementary treatise." 

Note B.p. 133. 

To account for some seeming discrepances, between the notes and 
text of this supplement, it is necessary to state that the text was drawn 
up in its present form from my manuscript, and the demonstration 
remodelled, (in many cases invented), to adapt it to the disolated state 
of the portions here given, during brief intervals stolen from othej 

• The figure may be easily sketched by the student. 

t A spherical four sided figure, whose sides axe all equal, and whose angles are also 
all equal. 

X A our sided spherical figure, or whose angles are equal ; or, perhaps, better adapted 
to the term, we may call it the figure in which great circles bisecting the pairs of oppo- 
site sides intersect each other at right angles. 

I' A figure whose opposite sides are equal. 

These terms are aaopt.^.^ by analogy from Plane Geometry. Perhaps it may be 
found desirable ere long to modify our terminology considerably : but it does not ap. 
pear to be the time. 



NOTES. 147 

pursuits and occupations, having but little alliance with these subjects. 
The notes were added afterwards, in a letter to Mr. Young, and dis- 
tributed by him so as not to interfere (where the interference would 
occasion much change in the text already partly in slips and partly in 
forms,) with the part already in the compositor's hands. Where addi- 
tion could be worked into the text, and appeared more adapted to 
incorporation, it has been done ; and where it did not coalesce with the 
text conveniently it has been put into foot notes. Some cases have, 
however, occurred where the addendum coald not be properly made 
by either method, and it has therefore been altogether omitted. Still 
as these omissions are rather of a historical than a mathematical 
nature, no inconvenience can result from them, except the possibly 
erroneous distribution of the names of discoverers of particular theo- 
rems. Should this be ultimately found to be the case, I trust the 
authors to whom they are erroneously attributed, as well as the authors 
to whom they are actually due, will excuse the undesigned mistake. 

There is, however, one particular case to which I wish more espe- 
cially to refer, since 1 had till just now considered a theorem upon 
which I set some value, (and which, indeed, was the origin of my 
researches upon these topics), to be original, when, in fact, it had been 
discovered more than a quarter of a century ago, by Professor Lowry. 
I have just received a note from that distinguished Geometer, con- 
taining, amongst other matters, a reference to the Mathematical Repo- 
sitory, N.s. vol. I. p. 157. 

Upon turning to this volume I find an anticipation of this beautiful 
property of the polar triangles : but as my copy of the Repository had 
been lent to a friend during the whole of the time I had the subject 
before my mind, my own discovery was perfectly independent of his, 
though so long posterior to it. I am quite sure, indeed, that I had 
never read that passage, or so beautiful a property must have been 
inevitably laid up amongst my collections. I am happy, however, to 
be able to render back to Professor Lowry the credit of the priority oi 
discovery in the same volume in which I had seemed to claim not 
only independence but priority. 

His demonstration (as was to be expected when the methods of sphe- 
rical research in general at the two periods are compared) differs 
totally from mine ; but his, as the geometrical often will have over the 
analytical, even when the latter is cultivated to its utmost perfection, 
has advantages over mine, which render it desirable to give it here. 
It is simple, and it proves more than mine proves, or perhaps can 
prove in moderate compass, viz. that the centres of the two circles, 
whose radii are complimentary, coalesce with one another. I will add, 
that to him alone we owe every important spherical theorem that can be 
set dovm to the credit of Englishvien during at least a century past, pro- 
bably even longer. 

Find the centre O of the inscribed a 

circle in ABC, and from the points of yT''^- 

contact G,H,K, draw the radii OG, OH, / /^ \ 

OK. Then these being perpendicular / // \ \ 

to the sides BC, CA, AB, respectively / /^ — >\ \ 

pass through the poles a, b, c, of those / ^^ I A \ 

sides. Hence by polar triangles, / /f \^ o^,-.-^-^\ 

«0G = iOH = cOK. =~. But / [X^"\\ j\ \ 

OG = OH = OK, and, therefore, Oa .•C'"^Cl_^V^\_^i^>^ I 

= Ob =z Oc, or O is also the centre of ^<^ Q^ >>tf 

the circle about the polar triangle abc : '■■"•- ^-^ 

that is, the centres of the primary in- — — ' 



148 



NOTES. 



scribed, and the polar circumscribed circles are coincident. In the 
same manner the centres if polar inscribed and primary circumscribed 
are coincident. And it has been shown that these radii are comple- 
mentary. 

I may remark that the expression for the distance of the inscribed 
and circumscribed centres, in terms of the radii themselves, has not yet 
been given. In piano that distance was so assigned by Mr. Landen, 
and has been very elegantly investigated by Mr. Lowry, in the Mathe- 
matical Repository, The" corresponding problem has been several 
times attempted, but other parts of the triangle have appeared in every 
result that has yet been published. The neatest form that I have seen is 
given anonymously in the Annales des Matamatiques, tom. vi. p, 223. viz. 

TV sin. a 4- sin. b -f sin. c . 

cos. u = — ' ' . sm. r cos. R. 

"^ sin. 5 sin. s — a sin. s — b sin. s — c 



THE END. 



MATHEMATICAL TABLES ; 

COMPREHENDING 

THE LOGARITHMS OF ALL NUMBERS 

FROM 1 TO 36,000; 

ALSO 

THE NATURAL AND LOGARITHMIC 

SINES AND TANGENTS; 

COMPUTED TO SEVEN. PLACES OF DECIMALS, AND ARRANGED 
ON AN IMPROVED PLAN ; 

WITH 

SEVERAL OTHER TABLES, 

USEFUL IN 

NAVIGATION AND NAUTICAL ASTRONOMY, 



OTHER DEPARTMENTS OF PRACTICAL MATHEMATICS 
BY J. R. YOUNG, 

AUTHOR OP "elements OF TRIGONOMETRY," &C. 



REVISED AND CORRECTED BY 

J. D. WILLIAMS, 

AUTHOR OP " key TO HUTTOn's MATHEMATICS," &C. 



PHILADELPHIA: 
CAREY, LEA, & BLANCHARD. 

1833. 



Entered according to the act of congress, in the year 1833, by Carey, 
Lea, & Blanchard, in the clerk's office of the district court of the eastern 
district of Pennsylvania. 



PREFACE TO THE TABLES. 



The following Tables are designed as well for the prac- 
tical man as for the mathematical student. They compre- 
hend, in a portable and cheap form, the principal informa- 
tion sought for in larger and more expensive collections. 

The more important of these tables, viz. those immedi- 
ately connected with trigonometrical and astronomical cal- 
culations, differ considerably both in form and arrangement 
from those jn general use ; and it is hoped that this de- 
parture from the usual plan, which has not been hastily 
made, will tend to increase the facility of reference. 

In the table of the Logarithms of Numbers a new device 
has been adopted to mark the change of figure, and the 
several columns are so printed that, in seeking for the 
number corresponding to any proposed logarithm, the lead- 
ing figures of the given logarithm may readily present them 
selves to the eye. Instead of omitting the several leading 
figures common to a number of successive logarithms, as is 
generally done, it has been recommended to preserve all the 
common figures, as at page 2 of these tables. This plan 
might perhaps facilitate, in a small measure, the writing out 
of a logarithm corresponding to a given number, but it 
would certainly render the detection of any given logarithm 
from among such a dense mass of figures much less easy. 

In the table of logarithmic sines and tangents, the trigo- 
nometrical lines are inserted to every second, for the two 
first and two last degrees of the quadrant, and the old ar- 
rangement is followed ; that is, the sines, cosines, &c. of the 
small arcs proceed in order from the top of the page to the 
bottom ; and those of the large arcs, complements of the 
former, proceed in the reverse order, from the bottom to the 
top. The bulk of the table, however, is arranged difterently ; 



VI PREFACE. 

the sines and tangents proceeding onwards to the end, 

and the cosines and cotano-ents in the reverse direction. 
This is the arrangement recommended by Professor Airy, in 
his Trigonometry, but it vv^as not originally my intention to 
adopt itT Its advantages, however, having been more clearly 
pointed out to me by some scientific friends, occupied in 
computmg the Nautical Almanack, and in the contmual 
use of trigonometrical tables, I have been induced to de- 
part from my first design, and to adopt the improved form. 
1 regret that I had not^come to this determination earlier, 
before the table for the two first degrees had been printed. 

The table of natural sines and tangents is arranged upon 
the same plan as the former. The remaining tables of the 
volume require no particular observations here : a more 
minute detail of particulars will be found in the introduc- 
torj,^ explanation preiixed. 

As accuracy in mathematical tables is of far more conse- 
quence than arrangement, it may be proper to state here that 
the present collection have all undergone very carefiil and 
repeated examination. The proofs of the tables of numbers, 
and of sines and tangents, were each compared twice with 
the tables of Bagay, Hutton, and Babbage, and in some cases 
with those of Taylor also ; and the impressions from the 
stereo t^qDed plates were again all compared with Hutton and 
Babbage. 2\Iany errors in Bagay"s Tables of nimibers were 
thus detected, and one or two in the last edition of Hutton ; 
in Mr. Babbage's table I could find no error, and I have no 
doubt they amply deserve the reputation for accuracy which 
thev have obtained. 

J. R. YOUNG, 
Jan. 1. 1S33. 



EXPLANATION OF THE TABLES. 



The principal tables in this collection are the three fol- 
lowing, 1st, a Table of the Common Logarithms of Num- 
bers ; 2d, a Table of the Logarithms of the Trigonometri- 
cal lines to radius 10^° ; and 3d, a Table of the natural nu- 
merical values of the same lines to radius unity. 

The explanation which we here propose to give of these 
tables must be understood to concern not the methods of 
computing them, but simply the manner of using them. 
The various methods of constructing a table of logarithms 
we have already discussed in a separate tract,* which will 
shortly be followed by a similar tract on the formation of a 
table of sines and tangents ; our object here, therefore, will 
be to explain the use of tables already constructed. 

TABLE L 

Of the Table of the Logarithm's of Numbers. 

The base of the system of Common Logarithms is 10 ; 
that is, every positive number is considered as some power, 
either whole or fractional, positive or negative, of the num- 
ber 10, audit is the exponent of this power which is called 
the logarithjn of the proposed number. If, therefore, we 
inquire what is the logarithm of any number, 60 for instance, 
we mean to ask what value the exponent x must have in order 
that 10^ may be equal to 60 ; the proper value, as far at least 
as seven places of decimals, is 1-7781513; that is to say 
I Qi 7781513^ 60. The method of ascertaining the proper value 
of x^ for any proposed number, is fully explained in our tract 
on logarithms above mentioned, but when the proposed 
number is any whole power of 10, whether positive or neg- 
ative, it will be immediately seen to be such by mere inspec- 
tion, and its logarithm will then be readily discovered. 
For example, the numbers. 

1, 10, 100, 1000, 10000, &c. 

* An Elementary Essay on the Computation of Logarithms. 



VIU EXPLANATION OF THE TABLES. 

are atonceseen to be positive powers of 10, which powers are 

10°, 10\ 10^ 10^ lOS &c. 
and the numbers / 

•],01,-001,-0001,&c. 
are as readily seen to be the following negative powers of 10, viz. 
10-S 10-^ 10-^ 10-^ &G. 
Hence, of the series of numbers 

....,10000,1000,100,10,1, -1, -01, -OOVOOOl, . . . . 
the logs, are 

...., 4, 3, 3, 1,0,-1,-2,-3,-4,.... 

All this is very obvious ; and it is further obvious that a 
number between any two terms of the first of these series 
will have its logarithm between the two corresponding terms 
of the second series. Thus the logarithm of a number be- 
tween 10 and 100 will lie between 1 and 2 ; in other words, 
the integral part of the logarithm of any number, consist- 
ing of but two integral places of figures, however many de- 
cimals may follow, will always be 1. 

In like manner, the logarithm of a number between 100 
and 1000 will be between 2 and 3, of a number between 
1000 and 10000 the logarithm will be between 3 and 4, and 
so on ; that is, when the proposed number has three places 
of integers the integral part of its logarithm will be 2, when 
the number has four places of integers the integral part of 
its logarithm will be 4, and generally when the number has 
n places of integers, the integral part of its logarithm will be 
n — 1 ; and this expresses the number of places which the 
highest denomination, or first figure in the proposed number, 
is from the unit's place. Thus if 24785* 37 be the number 
proposed, then, seeing that its first figure 2 is four places 
from the unit's place, we know that its log. is 4 -f a decimal. 
Upon the same principles the logarithm of any number 
between 1 and 1 is between — 1 and ; that is, it is — 1 + a 
decimal, the logarithm of any number between -01 and -1 is 
— 2 + a decimal, and generally the logarithm of any num- 
ber whose first significant figure is in the nth place of deci- 
mals is — u, and this expresses the number of places which 
the first significant figure in the proposed number is from 
the unit's place . Thus if -00000736 be the number pro- 
posed, we know that as the first significant figure 7 is six 
places from the unit's place, its logarithm must be — ^^ 6 + a 



1684-3 . . . . 


168-43 . . . 


16-843 . . . 


1-6843 . . . 


•16843 . . 


-016843 . . 


•0016843 . . 



EXPLANATION OF THE TABLES. IX 

decimal. Seeing, therefore, that the integral part of a lo- 
garithm is so easily found from the proposed number, it is 
thought sufficient to insert in the table only the decimal part ; 
accordingly, all the logarithms in a table of common logar- 
ithms must be understood to be decimals, although the de- 
cimal points may not appear.* 

A valuable peculiarity of the common system of loga- 
rithms or that whose base is 10, is this, viz. that the loga- 
rithms of all numbers consisting of the same significant 
figures differ only in their characteristics. For example, 
the log. of 16843 is 4-22641 94 

3-2264194 
2-2264194 
1-2264194 
0-2264194 
1-2264194 
2-2264194 
3-2264194 
&c. &c. 

That such must really be the case is very plain, for, as 

104-2264194 = 15843 .-. 103-2264194 = 1^1 == 1684-3, 
1022264194 = }^3 = 16843, &C. 

We may remark too, as a particular case of this property 
of the present system of logarithms, that the decimal part 
of the logarithm of a number consisting of any number of 
significant figures, either followed, or preceded, by ciphers, 
is always the same as if the ciphers were absent. Thus 
the decimal part of the logarithm of 358000 or of -00358 
or of 3580, (fee. is the same as the decimal part of the loga- 
rithm of 358, so that, in seeking for the decimal part of the 
logarithm of a proposed number in the table, we are to disre- 
gard the ciphers with which it may commence or terminate. 

Having stated these preliminary notions, we shall now 
enter more particularly into the manner of using the table 
of logarithms following. 

* In some few tables, however, the characteristics or integral parst 
of the logarithms are inserted, as well as the decimal parts. 

B 



X EXPLxlNATION OF THE TABLES. 

PROBLEM I. 

To find the logarithm of any number from 1 to 36000. 

If the proposed number either begin or end with ciphers 
these, as remarked above, are to be disregarded. The first 
significant figure to the right is to be considered as occu- 
pying the place of units, the preceding figures therefore 
will express so many tens. We must look for these leading 
figures in the column of tens in the table, and the horizon- 
tal row of logarithms against them will be that in which 
the sought logarithm occurs ; it will be found under that 
figure, printed in the Egyptian character, which agrees 
with the figure in the unit's place of the proposed number. 
This being premised, we shall proceed at once to a few ex- 
amples which will much better show the manner of using 
the table than any written direction. 

EXAMPLE I. 

Required the logarithm of 3265. 

The leading figures 326 of this number I find in the co- 
lumn marked tens^ at page 7 ; and carrying my eye along 
the horizontal row of logarithms, thus pointed out, I find 
in the vertical column headed 5 the logarithm sought, 
which (when the integral part 3 issuppHed) is 3-5138832 ; 
for the 38832 is considered to be preceded by the 51 a little 
above it. 

EXAMPLE II. 

Required the logarithm of 3266. 

The proper horizontal row of logarithms being found as 
before, I find that which is under the 6 to be 40162, which 
number is however considered to be preceded by the same 
figures as the number adjacent to it, or immediately before 
it, that is, by 51 ; hence supplying the index, or integral 
part, the required logarithm is 3'5140162. 

EXAMPLE III. 

Required the logarithm of 3236. 

Having found the horizontal row which contains the lo- 
garithm, by means of the 323 in the tens column, I find 
the part under the 6 to be 00085 which I should proceed to 



EXPLANATION OF THE TABLES. XI 

complete by prefixing, as in last example, the 50 belonging 
to the number immediately before it, were it not that the 
crooked mark/ directs me to the 51 helow^ so that, supply- 
ing the index, the required logarithm is 3-5100085. 

EXAMPLE IV. 

Required the logarithm of 4680000. 

Disregarding the terminating ciphers, I seek first for 46 
in the tens column, and I find it in page 2 of the table ; 
and in the same horizontal line with it, and under the 8, I 
find the decimal 6702459 ; hence, supplying the index, the 
required logarithm is 6-6702459. 

EXAMPLE V. 

Required the logarithm of -002138. 

Disregarding the ciphers, I seek first for 213 in the co- 
lumn of tens page 5, against which, and under the 8, 1 find 
the decimal 3300077 ; hence, prefixing the index, the re- 
quired logarithm is 3-3300077. 

PROBLEM II. 

To determine the logarithm of a number beyond the 
limits of the table. 

When the number proposed is beyond the limits of the 
table, that is, when it exceeds 30600. Enter the table 
with only the first five figures of the number, or indeed, 
with only the first four figures, should the five exceed the 
limits of the table, and find the corresponding logarithm. 
From the column marked dif. take out the number oppo- 
site to this logarithm, and multiply it by the remaining 
figures of the proposed number, reject from the product as 
many figures to the right as there are in the multiplier, and 
add the rest of the product to the logarithm already found : 
the sum will be the logarithm sought. 

EXAMPLE I. 

Required the logarithm of 843742. 

I first seek the logarithm of 8437, the four first figures, 
the five first being beyond the hmits of the table ; this lo- 
garithm I find at page 16 to be, without the index, 9261880, 



Xil EXPLANATION OF THE TABLES. 

and opposite to it in the column dif. is 515 ; this multipUed 
by 42, the remaining figures of the proposed number, 
produces 21630, from which product the two right-hand 
figures 30 being rejected, there remains 216 to be added to 
9261880, which gives 9262096 for the decimal part of the 
required logarithm ; therefore, prefixing the index 5, the 
complete logarithm is 5-9262096. 

EXAMPLE II. 

Required the logarithm of 1326927. 

Log. 132690 . . . 5-1228382 dif. 3277 

88 27 



Log. 1326927 . . . 5-1228470 2289 

654 



88,29 



EXAMPLE III. 

Required the logarithm of 114-1285. 

Log. 114-12 . . . 2-0573618 dif.ZSl 

324 85 



Log. 114-1285 . . . 2-0573942 1905 

3048 



323-85. 

It must be observed that as the colunm of differences does 
liot commence till page 7 of the table, the preceding pages 
are never to be consulted for the logarithm of a number be- 
yond the limits of the table. 

PROBLEM III. 

A logarithm being given, to find the corresponding 
number. 

In this problem, too, as in the last, reference will be made 
to those pages only which contain the dif. column ; among 
these we are to seek for the decimal part of the proposed 
logarithm, and we shall readily be guided to it, or else to a 
logarithm very near it, by means of the leading figures, 
which are separated in the table from the others, to attract 
the eye. If we find a logarithm exactly agreeing with 
that given, then the number, which the table shows us tp 



EXPLANATION OF THE TABLES. Xlll 

belong to the logarithm found, will be the required number. 
If, however, as is most likely, we do not find the proposed 
logarithm exactly, then we are to take out the number cor- 
responding to the next less logarithm ; this number will of 
course fall short of that required, but the deficiency may be 
supplied as follows. Divide the difference between the tab- 
ular logarithm and the given one by that number in the 
dif. column which is opposite to the tabular logarithm, and 
add the quotient to the number already taken from the table. 

EXAMPLE I. 

Required the number whose logarithm is 1-2335678. 

Given logarithm . . 2335678 

The next less in the table is log. 17122 . 2335545 



^^^ • 1 ^tab. dif 254)133-00 (-52 

Required number . 17-12252 1270 

~~600 

508 

~2 



EXAMPLE II. 

Required the number whose logarithm is 3-1241987. 
Given logarithm . ". . 1241987 
Next less log. 13309, . . 1241454 

163 



Tab. dif 326) 533-00(1-63 
Required number 1331-063 326 

2070 
1956 

1140 

These examples will, no doubt, be found sufficient to ex- 
amplify the manner of referring to the table when we are 
in search of a logarithm answering to a given number, or of 
a number answering to a given logarithm. We shall now 
give an example or two of the use of the table in facilitating 
arithmetical operations. 
2 



XIV EXPLANATION OF THE TABLES. 

PROBLEM IV. 

To multiply ?iumbers together. 
Add together the logarithms of the numbers, and the sum 
will be the loD;arithm of their product. 

EXAMPLE I. 

Required the product of 26784 and 7-865. 

log. 26784 . . 4-4278754 
log. 7-865 . . -8956987 

log. 210656-1 . . 5-3235741. 
hence the product is 210656-1. 

EXAMPLE II. 

Required the product of 3-586, 2-1046, -8372, and -0294. 

log. 3-586 . . .5546103 

2-1046 . . __.3231696 

.8372 . . 1-9228292 

.0294 . . 2^4683473 



Product -1857618 . . 1-2689564. 

PROBLEM V. 

To divide one number by another. 
Subtract the logarithm of the divisor from that of the div- 
idend, and the remainder will be the logarithm of the quo- 
tient. 

EXAMPLE I. 

Divide 28-654 by 127-34. 

log. 28-654 . . 1-4571853 
127-34 . . 2-1049648 



Quotient -2250197 . . 1-3522205. 

EXAMPLE II. 

Divide -06314 by -007241. 

log. -06314 . . ^-8003046 

auotient -007241 . . 3-8597985 

8-71979. . -9405061. 



EXPLANATION OP THE TABLES. XV 

PROBLEM VI. 

To find the nth power of a given number. 
The logarithm of the nih. power will be equal to n times 
the logarithm of the given number. 

EXAMPLE I. 

Required the fourth power of -09163. 



log. '09163 


2-9620377 
4 


Power -0000704938 . 


"5-8481508. 


EXAMPLE II. 




.lired the tenth power of -64. 
log. -64 


r8061800 
10 


:>ower -011529225 


2^0618000 



PROBLEM VII. 

To find the nth root of a giveri number. 
The logarithm of the nth. root will be equal to the nih. part 
of the logarithm of the given number. 

EXAMPLE I. 

Required the fourth root of -434296. 
log. -434296 . r6377858 

i of it . . . r9094464 = log. -811795 the root. 

As the negative index T, of the given logarithm, is not 

divisible by 4, it is increased by 3 to make it so, and the 3, 

thus borrowed, is afterwards restored, by being prefixed to 

the 6, making it 3*6 ; that is, the proposed logarithm is 

viewed under the form 4 + 3-6377858, to which it is obvi- 
ously equivalent. 

EXAMPLE II. 

Required the tenth root of 2. 
log. 2 . . -3010300 
Vo of it . . -0301030 = log. 1-000121 the root. 



XVI EXPLANATION OF THE TABLES. 

EXAMPLE III. 

Required the cube root of •00048. 
log. -00048 . 4-6812412 
i- of it . . 2-8937471 = log. -0782973 the root. 

The negative index 4 not being divisible by 3, it is in- 
creased by 2 to make it so, and then the borrowed 2 restored 
by considering the positive part to commence with 2-6 
instead of 6. 



TABLE II. 

Of the Table of Logarithmic Sines, Tangents, Cyc. 

This second table consists of two parts : the first part con- 
taining the logarithmic sines, cosines, &c. of the first two 
and of the last two degrees of the quadrant, computed to 
every single second ; and the other part of the table, contain- 
ing the trigonometrical lines of the intermediate part of 
the quadrant, for every minute only. 

The fijst part of the table, or that computed to seconds, is 
arranged in the usual manner ; that is, the sines, cosines, 
tangents, and cotangents of the small arcs proceed from 
the top of the page to the bottom, according to the magni- 
tude of the arcs, of which the degrees and minutes stand at 
the head of the columns, and the seconds occupy the left- 
hand column of every page. The sines, cosines, &c. of the 
large arcs, or those which are near 90°, and are the comple- 
ments of the former, proceed, on the contrary, from the bot- 
tom of the page to the top, according to the magnitude of 
the arcs, of which the degrees and minutes stand at the bot- 
tom, and the seconds occupy the right-hand column of 
every page. In entering this part of the table, therefore, with 
a small arc, the eye must be directed to the toip of the page, 
but on enterins: it with a larsre arc we must look to the hot- 
torn of the page. 

The arrangement of the remaining part of the table is 
different from that usually adopted ; for here the sines and 
tangents all proceed regularly, in the order of their magni- 
tudes, from the top to the bottom of the page ; while the cosines 



EXPLANATION OF THE TABLES. XVU 

and cotangents all proceed in the contrary order, that is, 
from the bottom of the page to the top. This arrangement 
has considerable advantages over that of other trigonometri- 
cal tables, of which we may mention the following as in- 
stances. Suppose we enter this table with an arc contain- 
ing seconds, as well as degrees and minutes, then if we seek 
its sine or tangent, that is, if we proceed down the table, the 
proportional difference, due to the seconds, will always be, 
additive ; but if we want the cosine or tangent, that is, if 
we proceed tip the table, then, on the contrarjr, the propor- 
tional difference will always be suhtractive. Again, sup- 
pose that we enter the table with a logarithmic line, in search 
of the corresponding arc. We may first find the nearest 
tabular value less than the proposed, note the correspond- 
ing degrees and minutes, and then proportion for the sec- 
onds, which will always be additive if we proceed down 
the table, that is, if the given line be a sign or tangent, and 
always subtractive if we proceed up the table, that is, if the 
given line be a cosine or cotangent ; of course the contrary 
will have place if we transcribe the nearest greater in- 
stead of the nearest less tabular value. But perhaps the 
principal advantage of the present arrangement is this, viz. 
that every opening of the table presents us with a greater 
number of consecutive sines, cosines, &c. than it could da 
under any other arrangement : and this peculiarity will 
always facilitate those operations which involve the sines, or 
the cosines, (fee. of several neighbouring arcs, (as in the lu- 
nar problem, for instance, where the true and apparent at- 
titudes of the bodies differ but little from each other.) The 
arc also, corresponding to any given logarithmic line, will 
be more readily found than under the old arrangement. 

We must remark here, that the secants and cosecants of 
arcs have not been inserted, because they may be immedi- 
ately supplied from the cosine and sines. For, since 

COS. : rad. :: rad. : sec. 

i'ad.2 , ^^ , 

.-. sec. = .'. loff. sec. f 20 — loff. cos. 

COS. ^ ^ 

and thus the log. secant of an arc is got by subtracting its 
log. consine from 20 ; and the log. consecant, by subtract- 
ing its log. sine from 20. Having spoken of the arrange- 
2* C 



XVlll EXPLANATION OF THE TABLES. 

merit of this table, we shall now more particularly describe 
the manner of referring to it. 

PROBLEM I. 

To find the log. sine, ^c. of a very small or of a very large 
arc, expressed in degrees, minutes, and seconds. 

By a very small arc we mean one not exceeding two de- 
grees ; and to find its log. sine, we first search among the 
left-hand pages of the early part of the table, for that which 
presents the proposed degrees and minutes at the top ; 
having found this, we shall have the vertical column in 
which the sine is ; we must then pass the eye down the 
left-hand column, till we come to the number of seconds, 
then, in the same horizontal line with this number, and, in 
the vertical column before found, we shall find the sine 
required. 

The tangent and cotangent are found in a similar man- 
ner among the right-hand pages. 

The same pages which contain the sines, cosines, &c. of 
arcs below 2°, contain also those of arcs above 88°. When 
such a large arc is given, we must seek for that page which 
presents the degrees and minutes of it at bottom, and we 
shall thus find the column in which the sought trigonome- 
trical line is ; the corresponding seconds column will be 
seen on the right of the page ; we must pass the eye up this 
till we reach the given number of seconds, opposite to which, 
in the vertical column already found, we shall see the 
sought number. 

To find the log. sine, log. cosine, 6^c. of an arc consisting of 
degrees and minutes only, and between 2° and 88° 

Within these limits the trigonometrical lines are given 
for every minute only ; but columns of differences are an- 
nexed, by means of which the proper correction for seconds 
may be easily found. 

In this part of the table the sines and tangents proceed 
throughout from the top to the bottom of the page ; the 
cosines and cotangents from the bottom to the top. If we 
enter the table with degrees and minutes, and seek for a 
sine, we look for the given degrees at the top of one of the 
left-hand pages ; if for a tangent, we look for the degrees at 



EXPLANATION OF THE TABLES. XIX 

the top of one of the right-hand pages : the minutes are to 
be found in the left-hand marginal column of the page : 
the number sought will be under the degrees at top, and in 
the same horizontal row as the minutes. But if we seek for 
a cosine or a cotangent, we look for the degrees at the bot- 
tom of the page instead of at the top, and for the minutes 
in the right-hand marginal column instead of in the left. 

To find the log. sine, 6(*c. when the arc consists of degrees, 
miiiutes, and seconds. 

In this case we enter the table with the degrees and 
minutes as before, and take out the corresponding number ; 
between this number and that which belongs to the succeed- 
ing minute we shall find, in the adjacent column, the proper 
di^erence. Multiply this difference by the number of 
seconds, divide the product by 60, and we shall have the 
correction to be applied to the tabular number : this correc- 
tion will be additive if we proceed down the table, or seek 
for a sine or tangent, but it will be subtractive if we proceed 
up, or look for a cosine or cotangent. We shall give an 
example or two of this operation. 

EXAMPLE I. 

Required the log. sine of 35° 27' 24''. 

Turning to page 126, we find for the log. sine of 35° 27' 
the number 9-764222, and the difference between this and 
the sine next following is shown in the difference column to 
be 1774, therefore the correction for 24" is 1774 X f 1 = 
1774 X -4 = 709-6, consequently, 

log. sin. 35° 27' = 9-7634222 
4- correction for 24" = 7096 

log. sin. 35° 27' 24" = 9-7634932 



EXAMPLE II. 

Required the log. cosine of 48° 35' 27". 
The log. cosine of 48° 35' we find, at page 128, to be 
9-8205496, therefore. 



XX EXPLANATION OF THE TABLES. 

loff. COS. 48° 35' = 9-820.5496 Dif. = 1433 

27 
— correction for 27" = 645 



10031 



9-8204851 2866 



6,0)3869,1 



645 = Cor. 



EXAMPLE III. 

Required the log. tangent of 15° 4d 31''. 

log. tan. 15° 43' = 9-4493260 Dif. = 4842 

H- correction for 31" = 2502 ^^ 



4842 



9-4495762 14526 



6,0)15010,2 



2501-7 = Cor. 



EXAMPLE IV. 

Required the log. cotangent of 41° 0' 29". 

log. cot. 41° 0' = 10-0608369 Dif = 2551 

— correction for 29" = 1233 



29 



22959 



log. cot. 41° 0' 29" = 10-0607136 5102 



6,0)7397,9 
1233 = Cor. 

EXAMPLE V. 

Required the log. secant of 13° 24' 23' . 
log. sec.l3°24'(=20 — log. cos.)=10-0119872. Dif.=301 

+ correction for 23" = 110 



23 



603 

log. sec. 13° 24' 23" = 10-0119982 602 

6,0)662^3 
nO=Ck>r, 



EXPLANATION OF THE TABLES. XXI 



EXAMPLE VI. 

Required the log. cosecant of 34° 52' 43''. 
log. cosec. 34° 52' ( = 20 — log. sin.) = 10-2428556.;..Dif. = 1812 
— correction for 43'' = 1299 



fi43fi 

log. cosec. 34° 52' 43" =. 10-2427257 ^^Is 



6,0)7791,6 

1298-6 = Cor. 

To find the arc corresponding to a given log. sine or log. 
tangent, ^'c. 

Search in the table for that log. sine, or log. tangent, 
which is nearest to the proposed, but less than it, and take 
out the corresponding degrees and minutes. Find also the 
diiFerence between this tabular number and the proposed, 
multiply it by 60 and divide by the tabular difference, the 
quotient will give the proper number of seconds. 

EXAMPLE I. 

Required the arc whose log. sine is 9-7634932. 
Given log. sine . 97634932 
log. sine 35° 27' . 9-7634222 tab. dif 1774 
24" 



710 

35°27'24"=req. arc 60 



1774)42600(24 
^3548 ~~ 



7120 
7096 



To find the arc corresponding to a given log, cosine or 
log. cotangent. 

Proceed, as in last problem, with this exception only, 
that instead of taking from the table the number next less, 
take the next greater ; or if we take the next less, we must 



subtract the correction, not add it. 



XXll EXPLANATION OF THE TABLES. 

EXAMPLE. 

Required the arc whose log. cosine is 9-8204851. 
Given log. cosine . . 9-8204851 
log. cos. 28° 35^ . . 9-8205496 tab. dif. 1433 

^^" ^45" 

Req. arc = 28° 35' 27" 60 



1433)38700(27 
2866 



10040 
10031 



TABLE III. 

Natural Sines, Tangents, S^c. 

This table is used like the former, but as the columns of 
differences are not inserted, when the difference between 
any two contiguous tabular numbers is required, for the 
purpose of correcting for seconds, this difference must be 
found by actual subtraction. 

TABLE IV. 

Traverse Table to every Quarter point of the Compass. 

This table is useful in Navigation, showing, by inspec- 
tion, the difference of latitude and departure due to any pro- 
posed course and distance. If the distance sailed be more 
than 120 miles it will exceed the limits of the table ; but 
the difference of latitude and departure may still be deter- 
mined from it by this simple operation : divide the given 
distance by any number that will give a quotient not exceed- 
ing 120; enter the table with this quotient, and multiply the 
corresponding dif of lat. and dep. by the assumed divisor, 
and there will result the dif of lat. and dep. due to the 
proposed distance. 

The construction of the traverse table is obvious ; the 
given distance and course being always the hypotenuse and 
adjacent angle from which the dif of lat. and dep. tabulated 
are computed. 



EXPLANATION OF THE TABLES. XXIU 

TABLE V. 

Workman s Table for correcting tJie Middle Latitude. 
This table is useful for correcting what in Navigation is 
called the }?iiddle latitude. It is usual, in middle latitude 
sailing, to consider the departure which a ship makes in 
sailing upon an oblique rhomb from one parallel of latitude 
to another, to be equal to the distance between the meridians 
left and come to, m.easured on the middle parallel (see Trig, 
p. 74-5) ; but, as this is not strictly accurate, a correction 
becomes necessary. This correction is furnished by the 
present table ; the given middle latitude is to be found in 
the first column to the left; in a horizontal line with which, 
and under the given difference of latitude, is inserted the 
proper correction to be added to the middle latitude to 
obtain the latitude in which the meridian distance is ac- 
curately equal to the departure. The formula for con- 
structing this table is obtained as follows : 
Let d = proper difF. of lat. 

D = meridional diff. of lat. 

m = middle latitude. 

M= m -f- correction. 

L = diff. of longitude. 

Then, (Trig. p. 75), tan. course =-^ ^ ^ ^ 

But, (Trig. p. 77), tan. course = ^^ 

cos. M X L rad. x L 
* d ^ D '* 

.'. correction = cos. 

TABLES VI., YIL, YIIL, IX., X., XL, and XIl. 

These are all tables of corrections to be applied to the ob- 
served altitudes of the celestial bodies ; the manner of using 
them must be sufficiently obvious from inspecting them, 
provided the object of the several corrections is clearly un 
derstood ; and this is explained at length in the chapter on 
Nautical Astronomy in the Trigonometry, where several 
examples of the corrections are given. 




LOGARITHMS OF NUMBEKS 



FROM 1 TO 36,000. 



LOGARITHMS OF NUMBERS FROM 1 TO 36,000. [Table I. 

Between 1 = log.'~i 0, and 600 = log -i 2-7781513. 



1 

0000000 
0413927 
3222193 
4913617 
6127839 
7075702 
7853298 
8512583 
9084850 
9590414 
0043214 
0453230 
0827854 
1172713 
1492191 
1789769 
2068259 
2329961 
2576786 
2810334 
3031961 
3242825 
3443923 
3636120 
3820170 
3996737 
4166405 
4329693 
4487063 
4638930 

4785665 
4927604 
5065050 
5198280 
5327544 
5453071 
5575072 
5693739 
5809250 
5921768 
6031444 
6138418 
6242821 
6344773 
6444386 
0541765 
6637009 
6730209 
6821451 
6910815 

6998377 
7084209 
7168377 
7250945 
7331973 
7411516 
7489629 
7566361 
7641761 
7715875 
1 



3 
3010300 
0791812 
3424227 
5051500 
6232493 
7160033 
7923917 
8573325 
9138139 
9637878 
0086002 
0492180 
0863598 
1205739 
1522883 
1818436 
2095150 
2355284 
2600714 
2333012 

3053514 
3263359 
3463530 
3654880 
3838154 
4014005 
4183013 
4345689 
4502491 
4653829 

4800069 
4941546 
5078559 
5211381 
5340261 
5465427 
55S7086 
5705429 
5820634 
5932861 

6042261 
6148972 
6253125 
6354837 
6454223 
6551384 
6646420 
6739420 
6830470 
6919851 
7007037 
7092700 
7176705 
7259116 
7339993 
7419391 
7497363 
7573960 
7649230 
7723217 
2 



3 

4771213 
1139434 
3617278 
5185139 
6334685 
7242759 
7993405 
8633229 
9190781 
9684829 
0128372 
0530784 
0899051 
1238516 



4 

6020600 
1461280 
3802112 
5314789 
6434527 
7323938 
8061800 
8692317 
9242793 
9731279 
0170333 
0569049 
0934217 
1271048 



» 1553360" 1583625 
184691411875207 
2121876 2148439 



2380461 
2624511 
2855573 
3074960 
3283796 
3483049 
3673559 
3856063 
4031205 
4199557 
4361626 
4517864 
4668676 

4814426 
4955443 
5092025 
5224442 
5352941 



240549 

2648178 

2878017 

3096002 

3304138 

3502480 

3692159 

3873898 

4048337 

4216039 

4377506 

4533183 

4683473 

4828736 

4969296 

5105450 

5237465 

5365584 



547774785490033 
5599066 5611014 



5717088 
5831988 
5943926 

6053050 
6159501 
6263404 
6364879 
6464037 
6560982 
6655810 
6748611 
6839471 
6928469 
7015680 
,7101174 
7185017 
7267272 



5728716 
5843312 
5954962 
6083814 
6170003 
6273659 
6374897 
6473830 
6570559 
8665180 
6757783 
6848454 
6937269 
7024305 
7109631 
7193313 
7275413 
7347998«7355989 
742725117435098 
7505084 7512791 
7581546 7589119 
7656686 7664128 
,7730547 7737864 
3 I 4 



5 I 

6989700 
1760913 
3979400 
5440680 
6532125' 
7403627 
8129134 
8750613 
9294189 
9777236 
0211893 
0606978 
0969100 
1303338 
1613680 
1903317 
2174839 
2430380 
2671717 
2900346 

3117539 
3324385 
3521825 
3710679 
3891661 
4065402 
4232459 
4393327 
4548449 
4698220 
4842998 
4983106 
5118834 
5250448 
5378191 
5502284 
5622929 
5740313 
5854607 
5965971 
6074550 
6180481 
6283889 
6384893 
6483600 
6580114 
6674530 
6766936 
6857417 
6946052 

7032914 
7118072 
7201593 
7283538 
7363965 
7442930 
7520484 
7596678 
7671559 
7745170 
5 



7781513 
2041200 
4149733 
5563025. 
6627578 
7481880 
8195439 



9344985 
9822712 

0253059 
0644580 
1003705 
1335389 
1643529 
1931246 
2201081 
2455127 
2695129 
2922561 

3138672 
334453S 
3541084 
3729120 
3909351 
4082400 
4248816 
4409091 
4563660 
4712917 



t 
8450980 
2304489 
4313638 
56820171 
6720979' 
7558749' 
8260748 
8864907 
9395193 
9867717 

0293838 
0681859 
1038037 
1367206 
1673173 
11958997 
82227165 
2479733 
2718416 
2944662 

3159703 
3364597 
3560259 
3747483 
3926970 
4099331 
4265113 
4424798 
4578819 
4727564 



4857214 

4996871 

5132176 

5263393 

5390761' 

5514500 

5634811 

5751878 

5865873 

5976952 

6085260 
6190933 
6294096 
6394865 
6493349 
6589648 
6683859 
6776070 
6866363 
6954817 

7041505 
7126497 
7209857 
7291648 
7371926 
7450748 
7528164 
7604225 
7678976 
7752463 
6 



4871384 
5010593 
5145478 
5276299 
5403295 
5526682 
5846661 
5763414 
5877110 
5987905 
6095944 
6201361 
6304279 
6404814 
6503075 
6599162 
6693169 
6785184 
6875290 
6963564 

7050080 
7134905 
7218106 
7299743 
7379873 
7458552 
7535831 
7611758 
7686381 
7759743 
T 



8 
9030900 
2552725 
4471580 
5797836 
6812412 
7634280 
8325089 
8920946 
9444827 
9912261 
0334238 
0718820 
11072100 
1398791 
1702617 
1986571 
2253093 
2504200 
2741578 
2968652 

3180633 
3384565 
3579348 
3765770 
3944517 
4116197 
4281348 
4440448 
4593925 
4742163 
4885507' 
5024271 
5158738 
5289167 
5415792 
5538830 
5658478 
5774918 
5888317 
5998831 
6106602 
6211763 
6314438 
6414741 
8512780 
8608655 
8702459 
6794279 
6884198 
6972293 

7058637 
7143298 
7226339 
7307823 
7387806 
7466342 
7543483 
7619278 
7693773 
7767012 
8 



9 

9542425 
2787536 
4623980 
5910846 
8901961 
7708520 
8388491 
8976271 
9493900 
9956352 
0374265 
0755470 
1105897 
1430148 
1731883 
2013971 
2278867 
2528530 
2764618 
2988531 

3201463 
3404441 
3598355 
3783979 
3961993 
4132998 
4297523 
4456042 
4608978 
4756712 

4899585 

5037907 

5171959 

530199 

5428254 

5550944 

5670284 

5786392 

5899496 

6009729 

6117233 

6222140 

6324573 

6424845 

6522463 

8618127 

6711728 

8803355 



6981005 
7067178 
7151674 
7234557 
7315888 
7395723 
7474118 
7551123 
7626786 
7701153 
7774268 
9 



Table i.] logaritphis of numbers from 1 to 36,000. 3 


Between GOO = log -^ 2.7781513, and 1200 = log "i 30791812. 

1 


tens. 


I 1 3 


13 -lis 6 1 1 » 


P 9 




60 


7783745 95955 


7803173 1036917817554 24726 


7831337 39036 


7846173 




CI 


7360412 67514 


74605 81884 


88751 95307 


7902852 09885 


7916906 




62 


7930916 37904 


7944830 51846 


795S3C0 65743 


72675 79596 


86506 




63 


8000294 07171 


8014037 20393 


8027737 34571 


8041394 48207 


8055009 




64 


63580 75350 


82110 88859 


95597/02325 
8162413' 69038 


8109043 15750 '8122447 




65 


S135S10 42476 


8149132 55777 


75654 82259 


88854 




68 


8202015 08580 


8215135 21681 


3223216 34742 


8241253 47765 


3254261 




67 


67225 73693 


80151 86599 


93033 99467 


3305887 12297 


8313698 




68 


8331471 37344 


8344207 50561 


8356906 63241 


69567 75834 


82192 




69 


94780 01061 


8407332 13595 


8419348 26092 


8432328 33554 


8444772 




70 


8457180/63371 

8518696' 24800 


69553 75727 


81891 88047 


94194/00333 
8555192' 61244 


8506462 




71 


8530895 36982 


8543060 49130 


67289 




72 


79353 85372 


91383 97386 


8603380 09366 


3615344 21314 


8627275 




73 


8639174 45111 


8651040 569618 62873 68778 


74675 80564 


86444 




74 


98182,04039 


8709883 15729"8721563 273S8 


3733206 39016 «8744813 




75 


8756399/62178 


67950 73713 


79470 85218 


90959 96692 


8302418 




76 


8813847 19550 


8825245 30934 


3336614 42288 


8847954 53612 


59263 




77 


70544 76173 


81795 87410 


93017 98617 


8904210 09796 


8915375 




78 


8926510 32068 


8937618 43161 


8948697 54225 


59747 65262 


70770 




79 


81765 87252 


92732 98205 


9003671 09131 


9014583 20029 


9025468 




80 


9036325 41744 


9047155 52560 


57959 63350 


68735 74114 


79485 




81 


90209 95560 


9100905 06244 


9111576 16902 


9122221 27533 


9132839 




82 


9143432 48718 


53998 59272 


64539 69800 


75055 80303 


85545 




83 


96010/01233 
9247960' 53121 


9203450 11661 


9216865 22063|9227255 32440 


9237620 




84 


58276 63424 


68567 737048 78334 83959' 89077 




85 


99296/04396 
9350032' 55073 


9309490 14579 


9319661 24738 


9329308 34873 


9339932 




86 


60103 65137 


70161 75179 


80191 85197 


90198 




87 


9400182 05165 


9410142 15114 


9420081 25041 


9429996 34945 


9439889 




88 


49759 54686 


59607 64523 


69433 74337 


79236 84130 


89018 




89 


98777>03649 
9547248' 52065 


9503515 13375 


9518230 23030 


9527924 32763 


9537597 




90 


56878 61684 


66486 71282 


76073 80856 


85639 




91 


95184 99948 


9604708 09462 


9614211 18955 


9623693 28427 


9633155 




92 


9642596 47309 


52017 56720 


61417 66110 


70797 75480 


80157 




93 


89497 94159 


98816/03469 
9745117' 49720 


9708116 12758 


9717396 22028 


9726656 




94 


9735896 40509 


54318 58911 


63500 68083 


72662 




95 


81805 86369 


90929 95484 9800034 04579 


9809119 13655 


9818186 




96 


9827234 31751 


9836263 40770 


45273 49771 


54265 58754 


63238 




97 


72192 76663 


81128 85590 


90046 94498 


98946/03389 
9943172' 47569 


9907827 




98 


9916690 21115 


9925535 29951 


9934362 38769 


51963 




99 


60737 65117 


69492 73364 


78231 82593 


86952 91305 


95655 




100 


0004341 08677 


0013009 17337 


0021661 25980 


0030295 34605 


0038912 




101 


47512 51805 


56094 60380 


64660 68937 


73210 77478 


81742 




102 


90257 94509 


98756/03000 
0141003' 45205 


0107239 11474 


0115704 19931 


0124154 




103 


0132587 36797 


49403 53593 


57788 61974 


66155 




104 


74507 78877 


82843 87005 


91163 95317 


99467/03613 
0240750^44857 


0207755 




105 


0216027 20157 


0224284 28406|0232525 36639 


48960 II 


106 


57154 61245 


65333 69416 73496 77572 


81644 85713 


89777 




107 


97895/01948 
0338257' 42273 


0305997 10043 0314085 18123 


0322157 26188 


0330214 




108 


46285 50293 


54297 58298 


62295 66289 


70279 




109 


78248 82226 86202 90173| 


94141 98106 


0402066 06023 
41476 45398 


0409977 




110 


0417873 21816 


0425755 29691 


0433623 37551 


49315 




111 


57141 61048 


64952 68852 


72749 76642 


80532 84418 


88301 




112 


96056 99929 


0503798 07663 


0511525 15384 


0519239 23091 


0526939 




113 


0534626 38464 


42299 46131 


49959 53783 


57605 61423 


65237 




114 


72856 76661 


80462 84260 


83055 91846 


95634 99419 


0603200 




; 115 


0610753 14525 


0618293 22058 


0625820 29578 0633334 37086 i 


40334 




i 116 


48322 52061 


55797 59530 


63259 66986 


70709 74428 


73145 




117 


85569 89276 


92980 96681 


0700379 04073 


0707765 11453 


0715138 




118 


0722499 26175 


0729847 33517 


37184 40847 


44507 48164 


51819 




119 


59118 62763 


66404 70043 


73679 77312 


80942 8456S 


88192 




1 5J \ 


3 4 


5 6 


•T 8 


9 


i 



4 LOGARITHMS OF NUMBERS FROM 1 TO 36,000. [Table I. 


Between 1200 = log "i 3.0791812, and 1800 = log -^ 3-2552725. 


tens. 


12 3 4^ 


5 6 


T 8 


9 




120 


0795430 99045 0802656 06265 


0809870 13473 


0817073 20669 


0824263 




1 


0831441 35026 38608 42187 


45763 49336 


52906 56473 


60037 




2 


67157 70712 74265 77814 


81361 84905 


88446 91984 


95519 




3 


0902581 06107 0909631 13152 


0916670 20185 


0923697 27206 


0930713 




4 


37718 41216 44711 48204 


51694 55180 


58665 62146 


65624 




5 


72573 76043 


79511 82975 


86437 89896 


93353 96806 


1000257 




6 


1007151 10594 


1014034 17471 


1020905 24337 


1027766 31193 


34616 




7 


41456 44871 


48284 51694 


55102 58507 


61909 65309 


68705 




8 


75491 78880 


82267 85650 


89031 92410 


95785 99159 


1102529 




9 


1109262 12625 


1115985 19343 


1122698 26050 


1129400 32747 


36092 




130 


42773 46110 


49444 52776 


56105 59432 


62756 66077 


69396 




1 


76027 79338 


82647 85954 


89258 92559 


95858, 99154 


1202448 




2 


1209028 12315 


1215598 18880 


1222159 25435 


1228709 31981 


35250 




3 


41781 45042 


48301 51558 


54813 58065 


61314 64561 


67806 




4 


74288 77525 


80760 83993 


87223 90451 


93676 96899 


1300119 




5 


1306553 09767 1312978 16187 


1319393 22597 


1325798 28998 


32195 




6 


38581 41771 44959 48144 


51327 54507 


57685 60861 


64034 




7 


70375 73541 


76705 79867 


83027 86184 


89339 92492 


95643 




8 


1401937 05080 


1408222 11361 


1414498 17632 


1420765 23895 


1427022 




9 


33271 36392 


39511 42628 


45742 48854 


51964 55072 


58177 




140 


64381 67480 


70577 73671 


76763 79853 


82941 86027 


89110 




1 


95270 98347 


1501422 04494 


1507564 10633 


1513699 16762 


1519824 




2 


1525941 28996 32049 35100 


38149 41195 


44240 47282 


50322 




3 


56396 59430 62462 65492 


68519 71544 


74568 77589 


80608 




4 


86640 89653 92663 95672 


98678/01683 
1628630^31614 


1604685 07686 


1610684 




5 


1616674 19666 


1622656 25644 


34596 37575 


40553 




6 


46502 49474 


52443 55411 


58376 61340 


64301 67261 


70218 




7 


76127 79078 


82027 84975 


87920 90864 


93805 96744 


99682 




8 


1705551 08482 


1711412 14339 


1717265 20188 


1723110 26029 


1728947 




9 


34776 37G88 


40598 43506 


46412 49316 


52218 55118 


56016 




150 63807 66699 


69590 72478 


75365 78250 


81133 84013 


86892 




1 


92645 95518 


98389/01259 
1826999^ 29850 


1804126 06992 


1809856 12718 


1815578 




2 


1821292 24147 


32698 35545 


38390 41234 


44075 




3 


49752 52588 


55422 58254 


61084 63912 


66739 69563 


72386 




4 


78026 80844 


83659 86473 


89285 92095 


94903 97710 


1900514 




5 


1906118 08917 


1911715 14510 


1917304 20096 


1922886 25675 


28461 




6 


34029 36810 


39590 42367 


45143 47918 


50690 53461 


56229 




7 


61762 64525 


67287 70047 


72806 75562 


78317 81070 


83821 




8 


89319 92065 


94809 97552 


2000293 03032 


2005769 08505 


2011239 




9 


2016702 19431 


2022158 24883 


27607 30329 


33049 35768 


38485 




160 


43913 46625 


49335 52044 


54750 57455 


60159 62860 


65560 




1 


70955 73650 


76344 79035 


81725 84414 


87100 89785 


92468 




2 


97830/00508 
2124540' 27202 


2103185 05860 


2108534 11205 


2113876 16544 


2119211 




3 


29862 32521 


35178 37833 


40487 43139 


45790 




4 


51086 53732 


56376 59018 


61659 64298 


66936 69572 


72207 




5 


77471 80100 


82729 85355 


87980 90603 


93225 95845 


98464 




6 


2203696 06310 


2208922 11533 


2214142 16750 


2219356 21960 


2224563 




7 


29764 32363 


34959 37555 


40148 42740 


45331 47920 


50507 




8 


55677 58260 


60841 63421 


65999 68576 


71151 73724 


76296 




9 


81436 84004 


86570 89134 


91697 94258 


96818 99377 


2301934 




170 


2307043 09596 


2312146 14696 


2317244 19790 


2322335 24879 


27421 




1 


32500 35038 


37574 40108 


42641 45173 


47703 50232 


52759 




2 


57809 60331 


62853 65373 


67891 70408 


72923 75437 


77950 




3 


82971 85479 


87986 90491 


92995 95497 


97998/00498 
2422929 25414 


2402996 




4 


2407988 10482 


2412974 15485 


2417954 20442 


27898 




5 


32861 35341 


37819 40296 


42771 45245 


47718 50189 


52658 




6 


57594 60059 


62523 64986 


67447 69907 


72365 74823 


77278 




7 


82186 84637 


87087 89536 


91984 94430 


96874 99318 


2501759 




8 


2506639 09077 


2511513 13949 


2516382 18815 


2521246 23675 


26103 




9 


30956 33380 


35803 38224 


40645 43063 


45481 47897 


50312 






1 2 


3 4 


5 6 


T 8 


9 






) Table i.] logarithms of numbers from 1 to 36,000. 5 jl 


Between 1800 = log. "^ 3-2552725, and 2400 = log.-i 3-38021 12. jj 


tens. I 1 2 


3 4: 


5 6 


T 8 


9 




180 


2555137 57548 


2559957 62365 


2564772 67177 


2569582 71934 


2574386 




1 


79185 815S2 


83978 86373 


88766 91158 


93549 95939 


98327 




2 


2603099 05484 


2607867 10248 


2612629 15008 


2617335 19762 


2622137 




3 


26883 29255 


31625 33993 


36361 38727 


41092 43455 


45817 




4 


50538 52896 


55253 57609 


59964 62317* 64669 67020 


69369 II 


5 


74064 76410 


78754 81097 


83439 35780 


88119 90457 


92794 




6 


97464 99797 


2702129 04459 


2706788 09116 


2711443 13769 


2716093 




7 


2720738 23058 


25378 27696 


30013 32328 


34643 36956 


39268 




8 


43888 46196 


48503 50809 


53114 55417 


57719 60020 


62320 




9 


66915 69211 


71506 73800 


76092 78383 


80673 82962 


85250 




190 


89821 92105 


94388 96669 


98950/01229 
2821633^23955 


2803507 05784 


2808059 




1 


2812607 14879 


2817150 19419 


26221 23486 


30750 




2 


35274 37534 


39793 42051 


44307 46563 


48817 51070 


53322 




3 


57823 60071 


62319 64565 


66810 69054 


71296 73538 


75778 




4 


80255 82492 


84723 86963 


89196 91428 


93660 95890 


98118^ 11 


5 


2902573 04798 


2907022 09246 


2911468 13689 


2915903 18127 


2920344 




6 


24776 26990 


29203 31415 


33626 35835 


38044 40251 


42457 




7 


46866 49069 


51271 53471 


55671 57869 


60067 62263 


64453 




8 


68845 71037 


73227 75417 


77605 79792 


81979 84164 


86348 




9 


90713 92893 


95073 97252 


99429 01605 
3021144^23309 


3003781 05955 


3008128 




200 


3012471 14641 


3016809 18977 


25474 27637 


29799 




1 


34121 36280 


38438 40595 


42751 44905 


47059 49212 


51363 




2 


55663 57812 


59959 62105 


64250 66394 


63537 70630 


72820 




3 


77099 79237 


81374 83509 


85644 87778 


89910 92042 


94172 




4 


98430/00557 
3119657^21774 


3102634 04809^3106933 09056'3111178 13300 "3115420' || 


5 


23889 26004 


28118 30231 32343 34454 


36563 




6 


40780 42887 


44992 47097 


49201 51303 53405 55505 


57605 




7 


61801 63898 


65993 68088 


70181 72273 74365 76455 


78545 




8 


82721 84807 


86893 88977 


91061 93143 95224 97305 


99384 




9 


3203540 05617 


3207692 09767 


3211340 13913 3215984 18055 


3220124 




210 


24261 26327 


23393 30457 


32521 34584 


36645 38706 


40766 




1 


44882 46939 


48995 51050 


53104 55157 


57209 59260 


61310 




2 


65407 67454 


69500 71545 


73589 75633 


77675 79716 


81757 




3 


85834 87872 


89909 91944 


93979 96012 


98045/00077 


3302108 




4 


3306167 08195 


3310222 12248 


3314273 16297 3313320' 20343' 22364 




5 


26404 28423 


30440 32457 


34473 36488 


38501 40514 


42526 




6 


46548 48557 


50565 52573 


54579 56585 


58539 60593 


62596 




7 


66598 68593 


70597 72595 


74593 76539 


73584 80579 


82572 




8 


86557 88547 


90537 92528 


94514 96502 


98438/00473 
341330r 20277 


3402458 




9 


3406424 08405 


3410386 12366 


3414345 16323 


22252 




220 


26200 28173 


30145 32116 


34086 36055 


38023 39991 


41957 




1 


45887 47851 


49314 51776 


53737 55698 


57657 59615 


61573 




2 


65486 67441 


69395 71343 


73300 75252 


77202 79152 


31100 




3 


84996 86942 


88337 90332 


92775 94713 


96660 98601 


3500541 




4 


3504419 06356 


3508293 10229 


3512163 14098 


3516031 17963 


19395 




5 


23755 25684 


27612 29539 


31465 33391 


35316 37239 


39162 




6 


43006 44926 


46846 48764 


50682 52599 


54515 56431 


58345 




7 


62171 64083 


65994 67905 


69314 71723 


73630 75537 


77443 




8 


81253 83156 


85059 86961 


88862 90762 


92662 94560 


96458 




9 


3600251 02146 


3604041 05934 


3607827 09719 


3611610 13500 


3615390 




230 


19166 21053 


22939 24825 


26709 28593 


30476 32358 


34239 




1 


37999 39878 


41756 43634 


45510 47336 


49260 51134 


53007 




2 


56751 58622 


60492 62361 


64230 66097 


67964 69830 


71695 




3 


75423 77285 


79147 81009 


82869 84728 


86587 88445 


90302 




4 


94014 95869 


97723 99576 3701428 03280' 


3705131 06931 


3708830 




5 


3712526 14373 


3716219 18065 


19909 21753 


23596 25433 


27279 




6 


30960 32799 


34637 36475 


38311 40147 


41933 43317 


45651 




7 


49316 51147 


52977 54807 


56636 58464 


60292 62119 


63944 




8 


67594 69418 


71240 73063 


74884 76704 


78524 30343 


82161 




9 


85796. 87612 


89427 91241 


93055 94868 


96680 98492 


3800302 




iSSS 


1 2 


3 4 


5 « 


T 8 


9 


J 



1* 



6 LOGARITHMS OF NUMBERS FROM 1 TO 36,000. [Table I. || 


Between 2400 = log.-i 3-3802112, and 3000 = log-^ 3-4771213. 11 


tens. 


1 2 


3 4 


5 « 


18 9 




240 


3803922 05730 


3807538 09345 


3811151 12956 


3814761 16565 3818368 




1 


21972 23773 


25573 27373 


29171 30969 


32767 34563 36359 




2 


39948 41741 


43534 45326 


47117 48908 


50698 52487 54275 




3 


57850 59636 


61421 63206 


64990 66773 


68555 70337 72118 




4' 75678 77457 


79235 81012" 82789 84565" 86340 88114* 898SS 




5 


93433 95205 


96975 98746 


3900515 02284|3904052 05819 


3907585 




6 


3911116 12880 


3914644 16407 


18169 19931 


21691 23452 


25211 




7 


28727 30485 


32241 33997 


35752 37506 


39260 41013 


42765 




8 


46268 48018 


49767 51516 


53264 55011 


56758 58504 


60249 




9 


63737 65480 


67223 68964 


70705 72446 


74185 75924 


77663 




250 


81137 82873 


84608 86343 


88077 89811 


91543 93275 


95007 




1 


98467/00196 


4001925 03653 


4005380 07106 


4008832 10557 


4012282 




2 4015728' 17451 


19173 20894 


22614 24333 


26052' 27771 


29488 




3 32921 34637 


36352 38066 


39780 41492 


43205 44916 


46627 




4^ 50047 51755 


53464 55171 


56878 58584 


60289 61994 


63698 




5 


67105 68807 


70508 72209 


73909 75608 


77307 79005 


80703 




6 


84096 85791 


87486 89180 


90874 92567 


94259 95950 


97641 




7 


4101021 02710 


4104398 06085 


4107772 09459 


4111144 12829 


4114513 




8 


17880 19562 


21244 22925 


24605 26285 


27964 29643 


31321 




9 


34674 36350 


38025 39700 


41374 43047 


44719 46391 


48063 




260 


51404 53073 


54742 56410 


58077 59744 


61410 63076 


64741 




1 


68069 69732 


71394 73056 


74717 76377 


78037 79696 


81355 




2 


84670 86327 


87983 89638 


91293 92947 


94601 96254 


97906 




3 


4201208 02859 


4204509 06158 


4207806 09454 


4211101 12748 


4214394 




4 17684 19328' 


20972 22615 24257 25898! 


27539 29180 


30820 




5 


34097 35735 


37372 39009 


40645 42281 


43916 45550 


47183 




6 


50449 52081 


53712 55342 


56972 58601 


60230 61858 


63486 




7 


66739 68365 


69990 71614 


73238 74861 


76484 78106 


79727 




8 


82968 84588 


86207 87825 


89443 91060 


92677 94293 


95908 




9 


99137 00751 
4315246^16853 


4302364 03976 


4305588 07199 


4308809 10419 


4312029 




270 


18460 20067 


21673 23278 


24883 26487 


28090 




1 


31295 32897 


34498 36098 


37698 39298 


40896 42495 


44092 




2 


47285 48881 


50476 52071 


53665 55259 


56851 58444 


60035 




3 


63217 64807 


66396 67985 


69573 71161 


72748 74334 


75920 




4 


79090 80675 


82258 83841 


85423 87005 


88587 90167 


91747 




5 


94906 96484 


98062 99639 


4401216 02792 


4404368 05943 


4407517 




6 


4410664 12237 


4413809 15380 


16951 18522 


20092 21661 


23230 




7 


26365 27932 


29499 31065 


32630 34195 


35759 37322 


38885 




8 


42010 43571 


45132 46692 


48252 49811 


51370 52928 


54485 




9 


57598 59154 


60709 62264 


63818 65372 


66925 68477 


70029 




230 


73131 74681 


76231 77780 


79329 80877 


82424 83971 


85517 




1 


88608 90153 


91697 93241 


94784 96327 


97868 99410 


4500951 




2 


4504031 05570 


4507109 08647 


4510185 11722 


4513258 14794 


16329 




3 


19399 20932 


22466 23998 


25531 27062 


28593 30124 


31654 




4 


34712 36241 


37769 39296 


40823 42349 


43875 45400 


46924 




5 


49972 51495 


53018 54540 


56061 575821 59102 60622 


62142 




6 


65179 66696 


68213 69730 


71246 72762 


74277 75791 


77305 




7 


80332 81844 


83356 84868 


86378 87889 


89399 90908 


92417 




8 


95433 96940 


98446 99953 


4601458 02963 


4604468 05972 


4607475 




9 


4610481 11983 


4613484 14985 


16486 17986 


19485 20984 


22482 




250 


25477 26974 


28470 29966 


31461 32956 


34450 35944 


37437 




1 


40422 41914 


43405 44895 


46386 47875 


49364 50853 


52341 




2 


55316 56802 


58288 59774 


61259 62743 


64227 65711 


67194 




3 


70158 71640 


73121 74601 


76081 77561 


79039 80518 


81996 




4 


84950 85427 


87903 89378 


90853 92327 


93801 95275 


96748 




5 


99692/01164 
4714384^ 15851 


4702634 04105 


4705575 07044 


4708513 09982 


4711450 




6 


17317 18782 


20-247 21711 


23175 24639 


26102 




7 


29027 30488 


31949 33410 


34870 36329 


37788 39247 


40705 




8 


436-20 45076 


46533 47988 


49443 50898 


52352 53806 


55259 




9 


58164 59616 


61067 62518 


63968 65418 


66867 68316 


69765 






1 2 


3 4 


5 6 


T S 


9 





Table i.] logarithms of numbers from 1 to 36,000. 


M 


Between 3000 = log -^ 3-4771213, and 3600 = log.-^ 3-5563025. 




tens. \ 2 1 


3 4 


5 6 


T 8 


9 


dif\ 


300 


4772660 74107 


4775553 76999 


477S445 79S90 


4781334 82778 


4784222 


1446 


1 


87103 88550 


89991 91432 


92873 94313 


95753 97192 


98631 


41 




4S01507 02945 


4804381 05818 


4807254 08689 


4810124-11559 


4812993 


36 


3 


15859 17292 


18724 20156 


21587 23018 


24448 25878 


27307 


31 


4 


3U164 31592 


33020 34446 


35873 37299 


38725 40150 41574* 


27 


5 


44422 45845 


47268 48690 


50112 51533 


52954 54375 


55795 


22 


6 


5S633 60052 


61470 62888 


64305 65722 


67138 68554 


69969 


17 


7 


72798 74212 


75626 77039 


78451 79863 


81275 82686 


84097 


12 


8 


86917 88326 


89735 91144 


92552 93959 


95366 96773 


98179 


08 


9 


4900990 02395 


4903799 05203 


4906607 08010 


4909412 10814 


4912216 


04 


310 


15018 16418 


17818 19217 


20616 22015 


23413 24810 


26207 


1399 


1 


29000 30396 


31791 33186 


34581 35974 


37368 38761 


40154 


95 


2 


42938 443-29 


45720 47110 


48500 49890 


51279 52667 


54056 


90 


3 


56831 58218 


59604 60990 


62375 63761 


65145 66529 


67913 


85 


4 


70679 72062 


73444 74825 


76206 77587 


78967 80347 81727- 


81 


5 


84484 85862 


87240 88617 


89994 91370 


92746 94121 


95496 


77 


6 


98245 99619 


5000992 02365 


5003737 05109 


5006481 07852 


5009222 


72 


7 


5011962 13332 


14701 16069 


17437 18805 


20172 21539 


22905 


68 


8 


25637 27002 


28366 29731 


31094 32458 


33821 35183 


36545 


63 


9 


39268 40629 


41989 43349 


44709 46068 


47426 48785 


50142 


59 


320 


52857 54213 


55569 56925 


58280 59635 


60990 62344 


63697 


55 


1 


66403 67755 


69107 70459 


71810 73160 


74511 75860 


77210 


51 


2 


79907 81255 


82603 83950 


85297 86644 


87990 89335 


90680 


47 


3 


93370 94714 


96057 97400 


98743/00085 
5112147' 13485 


5101427 02768 


5104109 


43 


4 


5106790 08130 


5109469 10808 


14823 16160 


17497 


39 


5 


20170 21505 


22841 24175 


25510 26844 


28178 29511 


30844 


35 


6 


33508 34840 


36171 37602 


38832 40162 


41491 42820 


44149 


30 


7 


46805 48133 


49460 50787 


52113 53439 


54764 56089 


57414 


26 


8 


60062 61386 


62709 64031 


65354 66676 


67997 69318 


70639 


22 


9 


73279 74598 


75917 77236 


78554 79872 


81189 82507 


83823 


18 


330 


86455 87771 


89086 90400 


91715 93028 


94342 95655 


96968 


14 


1 


99592/00903 
5212689' 13996 


5202214 03525 


5204835 06145 


5207455 08764 


5210073 


10 


2 


15303 16610 


17916 19222 


20528 21833 


23138 


06 


1 3 


25746 27050 


28353 29656 


30958 32260 


33562 34863 


36164 
49151 


02 


4 


38765 40064 


41364 42663 


43961 45259 


46557 47854 


1298 


5 


51744 53040 


54336 55631 


56925 58220 


59513 60807 


62100 


94 


6 


64685 65977 


67269 68560 


69851 71141 


72431 73721 


75010 


91 


7 


77588 78876 


80163 81451 


82738 84024 


85311 86596 


87882 


87 


8 


90452 91738 


93020 94304 


95587 96870 


98152 99434 


5300716 


83 


9 


5303278 04558 


5305839 07118 


5308398 09677 


5310955 12234 


13512 


80 


340 


16066 17343 


18619 19896 


21171 22446 


23721 24996 


26270 


76 


1 


28817 30090 


31363 32635 


33907 35179 


36450 37721 


38991 


72 


2 


41531 42800 


44069 45338 


46606 47874 


49141 50408 


51675 


68 


3 


54207 55473 


56738 58003 


59267 60532 


61795 63059 


64322 


64 


4 


66847 68109 


69370 70631 


71892 73153 


74413 75673 


76932 


61 


5 


79450 60708 


81966 83223 


84481 85737 


86994 88250 


89506 


58 


6 


92016 93271 


94525 95779 


97032 98286 


99538/00791 
5412047' 13296 


5402043 


55 


7 


5404546 05797 


5407048 08298 


5409548 10798 


14544 


51 


8 


17040 18288 


19535 20781 


22028 23274 


24519 25765 


27010 


47 


9 


29498 30742 


31986 33229 


34472 35714 


36956 38198 


39439 


43 


350 


41921 43161 


44401 45641 


46880 48119 


49358 50596 


51834 


39 


1 


54308 55545 


56781 58018 


59253 60489 


61724 62958 


64193 


35 


2 


66660 67894 


69126 70359 


71591 72823 


74055 75286 


76517 


32 


3 


78977 80207 


81436 82665 


83894 85123 


86351 87578 


88806 


29 


4 


91259 92486 


93712 94937 


96162 97387 


98612 99836 


5501060 


25 


5 


5503507 04730 


5505952 07174 5508396 09618 


5510839 12059 


13280 


22 


6 


15720 16939 


18158 19377 20595 21813 


23031 24248 


25465 


18 


7 


27899 29115 


30330 31545 32760 33975 


35189 36403 


37617 


15 


8 


40043 41256 


42468 43680 44892 46103 


47314 48524 


49735 


12' 


9 


52154 533631 54572 55781 S 56989 58197 


59404 60012 


61881 


J 


1 1 S i 3 4 1 5 6 


T 8 


9 



S LOGAEITHMS OF NUMBERS FROM 1 TO 36,000. [ Table TT* 




Between 3600 = log."^ 3-5563025, and 4200 = log.-i 36232493. | 




tens. 


i ^ 


3 4 


5 6 


t 8 


i> 


dif.\ 




360 


5564231 65437 


5566643 67848 


5569053 70257 


5571461 72665 


5573869 


12051 




1 


76275 77477 


7S680 79881 


81083 82284 


83485 84686 


85886 


2 




2 


83265 89484 


90683 91S82 


93080 94278 


95476 96673 


97870 


11981 




3 


56002t2 01458 


5602654 03849 


5605044 06239 


5607433 08627 


5609821 


5 




4 12207 13399 


14592 15784 16975 18167* 19358 20548' 


21739 


1 




5 


24118 25308 


26497 27685 


28874 3G062 31250 32437 


33624 


1189 




6 


35997 37183 


38369 39555 


40740 41925 43109 44293 


45477 


5 




7 


47844 49027 


50209 51392 


52573 53755 54936 56117 


57298 


1 




8 


59658 60838 


62017 63196 


64375 65553 


66731 67909 


69087 


1178 




9 


71440 72617 


73793 74969 


76144 77320 


78495 79669 


80843 


5 




370 


83191 84364 


85537 86710 


87882 89054 


90226 91397 


92568 


2 




1 


94910 96080 


97249 98419 


99588/00757 
5711263' 12429 


5701926 03094 


5704262 


1169 




2 


5706597 07764 


5708930 10097 


13594 14753 


15924 


6 




3 


18252 19416 


20580 21743 


22906 24069 25231 26393 
34518 35678' 36837 37996 


27555 


3 




4 29877 31038' 


32198 33358 


39154 







5 


41471 42628 


43786 44943 


46099 472561 48412 49568 
57650 58803 59956 61109 


50723 


1156 




6 


53033 54188 


55342 56496 


62261 


4 




7 


64565 65717 


66868 68019 


69170 70320 71470 72620 


73769 


1 




8 


76067 77215 


78363 79511 


80659 81806 S2953 84100 


85246 


1148 




9 


87538 88683 


89828 90973 


92118 93262 94406 95550 


96693 


5 




Isso 


98979/00121 
5810389^11529 


5801263 02405 


5803547 04688 5805829 06969 


5808110 


2 




1 1 


12668 13807 


14945 16084 17222 18359 


19497 


1138 




2 


21770 22907 


24043 25179 


26314 27450 28585 29719 


30854 


5 




3 


33122 34255 


35388 36521 


37654 38786 39918 41050 


42181 


3 




4 44443 45574 


46704 47834 


48963 50093 51222 52351 


53479 







5 


55735 56863 


57990 59117 


60244 61370 


62496 63622 


64748 


1127 




6 


66998 68123 


69247 70371 


71495 72618 


73742 74865 


75987 


4 




7 


78232 79353 


80475 81596 


82717 83838 


84958 86078 


87198 


1 




8 


89436 90555 


91674 927S2 


93910 95028 


96145 97263 


98379 


1118 




9 


5900612 01728 


5902S44 03959 


5905075 06189 


5907304 08418 


5909532 


6 




390 


11760 12873 


13986 15098 


16210 17322 


18434 19546 


20657 


2 




1 


22878 23988 


25098 26208 


27318 28427 


29536 30644 


31753 







2 


33968 35076 


36183 37290 


38397 39503 


40609 41715 


42820 


1107 




3 


45030 46135 


47239 48344 


49447 50551 


51654 52757 


53860 


4 




4 


56064 57166 


58268 59369 


60470 61571 


62671 63771 


64871 


1 




5 


67070 68169 


69268 70367 


71465 72563 


73661 74758 


75855 


1098 




6 


78048 79145 


80241 81336 


82432 83527 


84622 85717 


86811 


6 




7 


88999 90092 


91186 92279 


93371 94464 


95556 96648 


97739 


3 




8 


99922/01013 
6010S17' 11905 


6002103 03193 


6004283 05373 


6006462 07551 


6008640 







9 


12993 14081 


15168 16255 


17341 18428 


19514 


1087 




400 


21686 22771 


23856 24941 


26025 27109 


28193 29277 


30361 


4 




1 


32527 33609 


34692 35774 


36855 37937 


39018 40099 


41180 


1 




2 


43341 44421 


45500 465S0 


47659 48738 


49816 50895 


51973 


1079 




3 


54128 55205 


56282 57359 


58435 59512 


60587 61663 


62739 


6 




4 


64889 65963 


67037 68111 


69185 70259 


71332 72405 


73478 


4 




5 


75622 76694 


77766 78837 


70909 80979 


82050 83120 


84191 


2 




6 


66330 87399 


88468 89537 


90605 91674 


92742 93809 


94877 


1069 




7 


97011 98078 


99144/00210 
6109794^ 10857 


6101276 02342 


6103407 04472 


6105537 


6 




8 


6107666 08730 


11921 12984 


14046 15109 


16171 


4 




9 


18295 19356 


20417 21478 


22539 23599 


24660 25720 


26779 


r 




410 


28898 29957 


31015 32074 


33132 34189 


35247 36304 


37361 


1058 




1 


39475 40531 


41587 42643 


43698 44754 


45809 46863 


47918 


5 




2 


50026 51080 


52133 53187 


54240 55292 


56345 57397 


58449 


3 




3 


60552 61603 


62654 63705 


64755 65805 


66855 67905 


68954 







4 


71052 72101 


73149 74197 


75245 76293 


77340 78387 


79434 


1048 




5 


81527 82573 


83619 84665 


85710 86755i 87800 88845 


89889 


5 




1 6 


91977 93021 


94064 95107 


96150 97193 


98235 99277 


6200319 


3 




1 7 


6202402 03443 


6204484 05524 


6206565 07605 


6208645 09684 


10724 


1 




1 8 


12802 13840 


14879 15917 


16955 17992 


19030 20067 


21104 


1038 




9 


23177 24213 


25249 26284 


27320 28355 


29390 30424 


31459 


5 




IL 


1 -z 


3 4 


5 6 


1 8 


9 







Table i.] logarithms of numbers from 1 to 36,000. 9 


Between 4200 = log.-i 3-6232493, and 4800 = log.^i 3-6812412. 


tens.t 1 3|3 4|5 6|T 8 


9 


dif.\ 


420 


6233527 34560 


6235594 36627 


6237660 38693 


6239725 40757 


6241789 


1033! 


1 


43352 44334 


45915 46945 


47976 49006 


50036 51066 


52095 





2 


54154 55132 


56211 57239 


58267 59295 


60322 61350 


62377 


1028 


3 


64430 65457 


66483 67509 


68534 69560 


70585 71610 


72634 


5 


4 


74683 75707 


76730 77754 


78777 79800 


80823 81845 


82867 


3 


5 


84911 85933 


86954 87975 


88996 90016 


91037 92057 


93076 


1 


6 


95115 96134 


97153 98172 


99190/00209 
630936r 10377 


6301226 02244 


6303262 


1018 


7 


6305296 06312 


6307329 08345 


11393 12408 


13423 


6 


8 


15452 16467 


17481 18495 


19503 20522 


21535 22548 


23560 


3 


9 


25585 26597 


27609 28620 


29632 30643 


31654 32664 


33674 


1 


430 


35694 36704 


37713 38723 


39732 40740 


41749 42757 


43765 


1009 


1 


457S0 46788 


47795 48801 


49808 50814 


51820 52826 


53832 


6 


2 


55843 56848 


57852 58857 


59861 60865 


61869 62873 


63376 


4 


3 


65882 66884 


67887 68889 


69891 70893 


71894 72895 


73897 


2 


4 


75898 76898 


77898 78898 


79898 80897 


81896 82895 


83894 


1000 


5 


85891 86889 


87887 88884 


89882 90879 


91876 92872 


93869 


997 


6 


95861 96857 


97852 98847 


99842/00837 
640978r 10773 


6401832 02826 


6403820 


5 


7 


6405808 06802 


6407795 08788 


11765 12758 


13749 


3 


8 


15733 16724 


17715 18705 


19696 20686 


21676 22666 


23656 





9 


25634 26623 


27612 28601 


29589 30577 


31565 32552 


33540 


988 


440 


35514 36500 


37487 38473 


39459 40445 


41431 42416 


43401 


6 


1 


45371 46355 


47339 48323 


49307 50291 


51274 52257 


53240 


4 


2 


55205 56187 


57169 58151 


59133 60114 


61095 62076 


63057 


1 


3 


65018 65998 


66977 67957 


63936 69915 


70894 71873 


72851 


979 


4 


74803 75786 


76763 77741 


78718 79695* 80671 81648' 82624* 7|| 


5 


84576 85552 


66527 87502 


88477 89452 


90426 91401 


92375 


5 


6 


94322 95296 


96269 97242 


98215 99187 


6500160 01132 


6502104 


3 


7 


6504047 05018 


6505989 06960 


6507930 08901 


09871 10841 


11811 





8 


13749 14719 


15687 16656 


17624 18593 


19561 20528 


21496 


968 


9 


23431 24397 


25364 26331 


27297 28263 


29229 30195 


31160 


6 


450 


33090 34055 


35019 35984 


36948 37912 


38876 39839 


40802 


4 


1 


42728 43691 


44653 45616 


46578 47539 


48501 49462 


50423 


2 


2 


52345 53306 


54266 55226 


56186 57145 


58105 59064 


60023 





3 


61941 62899 


63857 64815 


65773 66730 


67683 68645 


69602 


958 


4 


71515 72471 


73427 74383 


75339 76294 


77250 78205 


79159 


5 


5 


81068 82023 


82977 83930 


84884 85837 


86790 87743 


83696 


3 


6 


90601 91553 


92505 93456 


94408 95359 


96310 97261 


98212, 


1 


7 


6600112 01062 


6602012 02962 


6603911 04860 


6605809 06758 


6607706 


949 


8 


09603 105-51 


11499 12446 


13393 14341 


15287 16234 


17181 


7 


9 


19073 20019 


20964 21910 


22855 23800 


24745 25690 


26634 


5 


460 


28522 29466 


30410 31353 


32296 33239 


34182 35125 


36067 


3 


1 


37951 38893 


39835 40776 


41717 42658 


43599 44539 


45480 


1 


2 


47360 48299 


49239 50178 


51117 52056 


52995 53934 


54872 


939 


3 


56748 57686 


58623 59560 


60497 61434 


62371 63307 


64244 


7 


4 


66116 67051 


67987 68922 


69857 70792 


71727 72661 


73595 


5 


5 


75463 76397 


77331 78264 


79197 80130 


81062 81995 


82927 


3 


6 


84791 85723 


86654 87585 


88516 89447 


90378 91308 


92239 


1 


7 


94099 95028 


95958 96887 


97816 98745 


99674/00602 
6708950' 09876 


6701530 


929 


8 


6703386 04314 


6705242 06169 


6707096 08023 


10802 


7 


9 


12654 13580 


14506 15431 


16356 17281 


18206 19130 


20054 


5 


470 


21903 22826 


23750 24673 


25596 26519 


27442 25365 


29287 


3 


1 


31131 32053 


32974 33896 


34817 35738 


36659 37579 


38500 


I 


2 


40340 41260 


42179 43099 


44018 44937 


45856 46775 


47693 


919 


3 


49529 50447 


51365 52283 


53200 54117 


55034 55951 


56867 


7 


4 


58700 59615 


60531 61447 


62362 63277 


64192 65107 


66022 


5 


5 


67850 68764 


69678 70592 71505 72418 


73332 74244 


75157 


3 


6 


76982 77894 


78806 79718 80629 81540 


82452 83362 


84273 


1 


7 


86094 87004 


87914 88324 89734 90643 


91552 92461 


93370 


909 


8 


95187 96096 


97004 97912 98819 99727 


6800634 01541 


6802448 


8 


9 


6804262 05168 


6806074 06980 6807886 08792 09697 106021 


11507 


6 




1 25 


3 ^ 5 SIT 8198 Ij 



10 LOGAEITHMS OF NUMBERS FROM 1 TO 36,000. [Table 1. 

Between 4800 = log "i 3-6812412, and 5400 = log.-i 3-7323938. 



tens. 

480 
1 
2 
3 
4 
5 
6 
7 
8 
9 

490 
1 
2 
3 
4 
5 
6 



500 
1 
2 
3 
4 
5 
6 
7 
8 
9 

510 
1 
2 
3 
4 
5 
6 
7 



530 
1 
2 
3 
4 
5 
6 



1 J^ 

6813317 14222 
22354 23256 
31371 32272 
40370 41269 
49351 50248 
53313 59208 
67256 68150 
76181 77073 
85088 85978 
93977 94864 



6902847 
11699 
20534 
29350 
38149 
46929 
55692 
64438 
73165 
81876 



03733 
12584 
21416 
30231 
39027 
47806 
56568 
65311 
74037 
82746 



90569 91437 
99244/00111 
7007902^08767 
16543 17406 
25167 26028 
33774 34633 
42363 43221 
50936 51792 
59492 60347 
68031 68884 

76553 77405 
85059 85908 
93543 94396 
7102020 02866 
10476 11321 
18915 19759 
27339 28180 
35745 36585 
44136 44974 
52510 53347 



60869 
69211 
77537 

85847 
94142 

7202420 
10683 
18930 
27162 
3537S 
43578 
51763 
59933 
68087 
76226 
84350 
92458 

7300552 



3 

6815126 
24159 
33173 
4216S 
51145 
60103 
69043 
77964 
86857 
95752 

6904619 
13468 
22298 
31111 
39906 
43683 
57443 
66185 
74909 
83616 
92305 

7000977 
C9632 
18269 
26890 
35493 
44079 
52649 
61201 
69737 
78256 
86753 
95244 

7103713 
12165 
20601 
29021 
37425 
45812 
54183 



4: 

16030 
25061 
34073 
43066 
52041 
60998 
69936 
78355 
87757 
96640 

05505 
14352 
23130 
31991 

40785 
49560 

53318 
67053 
75780 
84485 

93173 
01843 
10496 
19132 
27751 
36352 
44937 
53505 
62055 
70589 
79107 
87607 
96091 
04559 
13010 
21444 
29862' 
38264 
46650 
55019 



16693 
1 



61703 

70044 
78369 
86677 
94970 
03247 
11508 
19754 
2T984 
36198 

44397 
52581 
60749 
68901 
770391 
85161 
93268 
01360 
09437 
17499 
2 



62538 63373 
70377 71710 
79200 80032 
87507 83337 
95799 96627 

7204074 04901 
12334 13159 
20578 21401 
28806 29628 
37019 37839 
45216 46035 
53398 54216 
61565 62380 
69716 70530 
77852 78^64 
85972 86784 
94078 94888 

7302168 02977 
10244 11051 
18304 19109 
3 4 



t 8 

6818741 19645 
27766 23668 
36773 37673 
- 45761 46659 
54730 55626 
63681 64575 
72613 73506 
61528 82418 
90423 91312 
99301-00188 
690S16r09046 
17002 17885 
25826 26707 
34631 35511 
43419 44297 
52189 53065 
60942 61816 
69676 70549 
78394 79264 
87093 87963 



5 6 

6816934 17333 

25963 26865 

34973 35873 

43965 44863 

52938 53834 

61392 62737 

70823 71721 

79746 80637 

88646 89535 

97527 98414 
6906390 07275 

15235 16119 

24062 24944 

32372 33752 

41663 42541 

50437 51313 

59193 60067 

67931 68804 

76652 77523 

85355 66224 

94041 94908 
7002709 03575 

11361 12225 

19995 20357 

28612 29472 

37212 38071 

45794 46652 

54360 55216 

62910 63764 

71442 72294 

79957 80808 

88456 89305 

96939 97786 
7105404 06250 

13854 14698 

22287 23129 

30703 31544 

39104 39943 

47483 48325 

55356 56691 

64207 65042 

72543 73376 

80363 81694 

89167 89996 

97455 93233 
7205727 06554 

13984 14309 

22225 23043 

30450 31272 

38660 39480 

46854 

55033 

63196 

71344 

79477 

87595 884061 89216 

95697 """^" '""'''' 
7303785 

11857 

19914 
S 



47672 
55850 
64012 
72158 
60290 



9650' 
04593 
12663 
20719 



95776 
7004441 
13089 
21720 
30333 
38930 
47509 
56072 
64617 
73146 

81659 
90154 
98633 

7107096 
15542 
23971 
32385 
40782 
49162 
57527 
65876 
74208 
82525 
90826 
99111 

7207330 
15633 
23871 
32093 
40300 

48491 
56667 
64827 
72972 
81102 



97316 

7305400 

13470 

21524 

K 



96643 
05307 
13953 
22582 
31193 
39788 
48366 
56927 
65471 
73996 

82509 
91003 
99480 
07941 
16385 
24813 
33225 
41620 
50000 
53363 
66710 
75041 
83356 
91655 
99938 
08206 
16458 
24694 
32914 
41120 

49309 
57483 
65642 
73786 
81914 
90027 
93125 
06208 
14276 
22329 
8 



9 

6820548 
29569 
38572 
47556 
56522 
65469 
74393 
83308 
92200 

6901074 

09930 
18768 
27583 
36390 
45175 
53941 
62690 
71421 
80135 
88831 

97510 

7006172 
14816 
23444 
32054 
40647 
49223 
57782 
66325 
74850 
83359 
91851 

7100327 
08786 
17229 
25655 
34065 
42459 
50837 
59198 
67544 
75873 
84186 
92464 

7200766 
09032 
17282 
25517 
33736 
41939 
50127 
58300 
66457 
74599 
82726 
90638 
98934 

7307015 
15082 
23133 
9 



dif. 

904 
2 


898 
6 
5 
3 
1 

889 
7 
5 
3 
2 
1 

879 
7 
5 
3 
2 


868 
6 

I 

l\ 

656! 
6 
5 
3 
1 

819 
6 
6 
4 
3 
1 


838 
7 

5 
3 
1 


826 
6 
5 
4 
2 
1 

819 

7 
6 

4 
3 

1 
809 

8 

t 



Table I.] LOGAKITHMS OF NU:MBEKS FROM 1 TO 36,000. 11 jj 


Between 5400 = log."' 3-7323938, and 6000 = log -^ 3-7781513. | 


tens. 


1 3 3 4|5 6 T 8191 


d^f-l 


54U 


7324742 25546 7326350 27153 7327957 28760 7329564 30367l7331170| 


804 1 


1 


32775 33578 34:^30 35183 351JS5 36787 


37533 38390 39192 
45598 46398^ 47198] 


21 


2 


407:U 41595 423S6 43197 43997 44798 


ol 


3 


4S798 495S8 50397 51196 519J5 52794 


53593 54392 


55191 


799 


4^ 


56787 575S5 58383 59181 > 59979 60776^ 


61574 62371 


63168^ 


8 


5 


64762 65558 66355 671511 


67943 G8744 


69540 70335 


71131 


7 


6 


72722 73617 74312 751071 


75902 76696 


77491 78285 


79079 


5 


7 


60667 81461 82254 83048 


83841 84634 


65427 86220 


87013 


4 


8 


86593 89390 90182 90974 


91766 92558 


93350 94141 
7401257 02047 


94932 


2 ' 


9 


96514 97305 9S096 98887 


99677 00467 


7402837 


Ol 


550 


7404416 052G6 7405995 06784 


7407573/08362 


09151 09939 


10728 


7891 


1 


12304 13092 13680 14668 


15455 16243 


17030 17817 18604 


8 


2 


20177 20964 21750 22537 


23323 24109 


24895 25680 26466 


6 


3 


28037 28822 29607 30392 


31178 31961 


32745 33530 34314 


5 


4 


35882 36665 37449 38232 


39016 39799 


40582 41365 42147 


4 


5 


43712 44495 


45277 46059 


46841 47622 


48404 49185 


49967 


2 


6 


51529 52310 


53091 53871 


54652 55432 


56212 56992 


57772 


1 


7 


59332 60111 


60890 61670 


62449 63228 


64006 64785 


65564 


779 


8 


67120 67898 


68676 69454 


70232 71009 


71787 72564 


73341 


8 


9 


74895 75672 


76448 77225 


78001 78777 


79553 80329 


61105 


7 


560 


82656 83431 


84206 84981 


85756 86531 


87306 88080 


88854 


5 


1 


90403 91177 


91950 92724 


93498 94271 


95044 95817 


96590 


4 


2 


98136 98908 


99681/00453 
7507398' 08168 


7501225 01997 


7502769 03541 


7504312 


2 


3 


7505855 06626 


08939 09710 


10480 11251 


12021 


1 


4 


13561 14331 


15101 15870 


16639 17409 


18178 18947 


19716 





5 


21253 22022 


22790 23558 


24326 25094 


25862 26629 


27397 


768 


6 


28932 29699 


30466 31232 


31999 32766 


33532 34298 


35065 


7 


7 


36596 37362 


38128 38893 


39659 40424 


41189 41954 


42719 


6 


8 


44248 45012 


45777 46541 


47305 48069 


48832 49596 


50359 


4 


9 


51886 52649 


53412 54175 


54937 55700 


56462 57224 


57987 


3 


570 


59510 60272 


61034 61795 


62556 63318 


64079 64840 


65600 


2 


1 


67122 67882 


68642 69402 


70162 70922 


71682 72442 


73201 





2 


74719 75479 


76237 76996 


77755 78513 


79272 80030 


80788 


759 


3 


82304 83062 


83819 84577 


85334 86091 


86848 87605 


88362 


7 


4 


89875 90632 


91388 92144 


92900 93656 


94412 95168 


95923 


6 


5 


97434 98189 


98944 99699 


7600453 01208 


7601962 02717 


7603471 


4 


6 


7604979 05733 


7606486 07240 


07993 08746 


09500 10253 


11005 


3 


7 


12511 13263 


14016 14768 


15520 16272 


17024 17775 


18527 


2 


8 


20030 20781 


21532 22283 


23034 23784 


24535 25285 


26035 


1 


9 


27536 28286 


29035 29785 


30534 31284 


32033 327821 33531 
39518 40266 41014 


749 


580 


35029 35777 


36526 37274 


38022 38770 


8 


1 


42509 43256 


44003 44750 


45497 46244 


46991 47737 


48484 


7 


2 


49976 50722 


51468 52214 


52959 53705 


54450 55195 


55941 


6 


3 


57430 58175 


58920 59664 


60409 61153 


61897 62641 


63385 


5 


4 


64872 65616 


66359 67102 


67845 68588 


69331 70074 


70816 


3 


5 


72301 73043 


73785 74527 


75269 76011 


76752 77494 


78235 


2 


6 


79717 80458 


81199 81940 


82680 83421 


84161 84901 


85641 





7 


87121 87860 


88600 89339 


90079 90818 


91557 92296 


93035 


739 


8 


94512 95250 


95988 96727 


97465 98203 


98940 99678 


7700416 


8 


9 


7701890 02627 


7703364 04101 


7704838 05575 


7706311 07048 


07784 


7 


590 


09256 09992 


10728 11463 


12199 12934 


13670 14405 


15140 


6 


1 


16610 17344 


18079 18813 


19547 20282 


21016 21750 


22483 


4 


2 


23951 24684 


25417 26150 


26884 27616 


28349 29082 


29815 


3 


3 


31279 32011 


32743 33475 


34207 34939 


35670 36402 


37133 


2 


4 


38596 39326 


40057 40788 


41519 42249 


42979 43710 


44440 


1 


5 


45900 46629 


47359 48088 


48818 49547 


50276 51005 


51734 





6 


53191 53920 


54648 55376 


56104 56832 


57560 58288 


59016 


728 


7 


60471 61198 


61925 62652 


63379 64106 


64833 65559 


66286 


7 


8 


67738 68464 


69190 69916 


70642 71367 


72093 72818 


73543 


6 


9 


74993 75718 


76443 77167 


77892 78616 


79340 80065 


80789 


5 




1 2 


3 4 


5 6 


T 8 


9 





12 LOGARITHMS OF NUMBERS FROM 1 TO 36,000. 


{Table I A 


Between 6000 = log."i 3-7781513, and 6600 = log-.~i 3-8195439. 




tens. 


1 s 


3 4: 6 6 


T 8 


I 9 


dif. 


600 


7782236 82960 


7783683 84407 7785130 85853 


7786576 87299 


7738022 


723 


1 


89467 90190 


90912 91634 92356 93078 


93800 94522 


95243 


2 


2 


96686 97408 


98129 98850 99571 00291 


7801012 01732 


7802453 


1 


3 


7803893 04613 


7805333 06053 7806773/07492 


08212 08931 


09650 





4 


11088 11807 


12526 13245 13963 14681 


15400 16118 


16836 


718 


5 


18272 18989 


19707 20424 


21141 21859 


22576 23293 


24010 


7 


6 


25443 26159 


26876 27592 


28308 29024 


29740 30456 


31171 


6 


7 


32602 33318 


34033 34748 


35483 36178 


36892 37607 


38321 


5 


8 


39750 40464 


41178 41892 


42606 43319 


44033 44746 


45460 


4 


9 


46886 47599 


48312 49024 


49737 50450 


51162 51874 


52586 


3 


610 


54010 54722 


55434 56145 


56857 57568 


58279 58990 


59701 


2 


1 


61123 61833 


62544 63254 


63965 64675 


65385' 66095 


66805 


1 


2 


68224 68933 


69643 70352 


71061 71770 


72479 73188 


73896 


709 


3 


75313 76021 


76730 77438 


78146 78854 


79561 80269 


80976 


8 


4 


82391 83098 


83805 84512 


85219 85926 


86632 87339 


88045 


7 


5 


89457 90163 


90869 91575 


92281 92986 


93692 94397 


95102 


6 


6 


96512 97217 


97922 98626 


99331 ,00035 


7900739 01444 


7902148 


5 


7 


7903555 04259 


7904963 05666 


7906370/07073 


07776 08479 


09182 


4 


8 


10587 11290 


11992 12695 


13397 14099 


14801 15503 


16205 


2 


9 


17608 18309 


19011 19712 


20413 21114 


21815 22516 


23216 


1 


620 


24617 25318 


26018 26718 


27418 28118 


28817 29517 


30217 





1 


31615 32314 


33014 33712 


34411 35110 


35809 36507 


37206 


699 


2 


38602 39300 


39998 40696 


41394 42091 


42789 43486 


44183 


8 


3 


45578 46274 


46971 47668 


48365 49061 


49757 50454 


51150 


7 


4 


52542 53238 


53933 54629 


55324 56020 


56715 57410 


58105 


6 


5 


59495 60190 


60884 61579 


62273 62967 


63662 64356 


65050 


5 


6 


66437 67131 


67824 68517 


69211 69904 


70597 71290 


71983 


4 


7 


73368 74060 


74753 75445 


76137 76829 


77521 78213 


78905 


2 


8 


80288 80979 


81671 82362 


83053 83744 


84435 85125 


85816 


1 


9 


87197 87887 


88577 89267 


89957 90647 


91337 9202? 


92716 





630 


94095 94784 


95473 96162 


96851 97540 


98228 98917 


99605 


689 


1 


8000982 01670 


8002358 03046 


8003734 04421 


8005109 05796 


8006484 


8 


2 


07858 08545 


09232 09919 


10605 11292 


11978 12665 


13351 


7 


3 


14723 15409 


16095 16781 


17466 18152 


18837 19522 


20208 


5 


4 


21578 22262 


22947 23632 


24316 25001 


25685 26369 


27053 


4 


5 


28421 29105 


29789 30472 


31156 31839 


32522 33205 


33888 


3 


6 


35254 35937 


36619 37302 


37984 38666 


39348 40031 


40712 


2 


7 


42076 42758 


43439 44121 


44802 45483 


46164 46845 


47526 


1 


8 


48887 49568 


50248 50929 


51609 52289 


52969 53649 


54329 





9 


55688 56368 


57047 57726 


58405 59085 


59764 60442 


61121 


679 


640 


62478 63157 


63835 64513 


65191 65869 


66547 67225 


67903 


8 


1 


69258 69935 


70612 71290 


71967 72644 


73320 73997 


74674 


7 


2 


76027 76703 


77379 78055 


78731 79407 


80083 80759 


81434 


6 


3 


82785 83460 


84136 84811 


85486 86160 


86835 87510 


88184 


5 


4 


89533 90207 


90881 91555 


92229 92903 


93577 94250 


94924 


4 


5 


96270 96944 


97617 98290 


98962 99635 


8100308 00980 


8101653 


3 


6 


8102997 03670 


8104342 05013 


8105685 06357 


07029 07700 


08372 


2 


7 


09714 10385 


11056 11727 


12398 13068 


13739 14409 


15080 


1 


8 


16420 17090 


17760 18430 


19100 19769 


20439 21108 


21778 





9 


23116 23785 


24454 25123 


25792 26460 


27129 27797 


28465 


669 


650 


29802 30470 


31138 31805 


32473 33141 


33808 34475 


35143 


8 


1 


36477 37144 


37811 38478 


39144 39811 


40477 41144 


41817 


7 


2 


43142 43808 


44474 45140 


45805 46471 


47136 47801 


48463 


6 


3 


49797 50462 


51127 51791 


52456 53120 


53785 54449 


55110 


5 


4 


^6441 57105 
^3076 63739 


57769 58433 


59097 59760 


60423 61087 


61750 


4 


5 


64402 65064 


65727 66389 
72347 73009 


67052 67714 


68376 


3 


6 


69700 70362 


71024 71686 


73670 74331 


74993 


2 


7 


76315 76976 


77636 78297 


78958 79618 


80278 80939 


81599 


1 


8 


82919 83579 


84239 84898 


85558 86217 


86877 87536 


88195 





L 


89513 90172 


90831 91489 


92148 92806 


93465 94123 


94781 


659 


1 2 


3 4 


5 6 


T 8 


9 





Table i.] logarithms of numbers from 1 to 36,000. 

Between 6600 ^ log.-i 38 195439, and 7200 = log.-' 3-8573325. 



13 



tens. 
660 
1 
2 
3 
4 
5 
6 
7 
8 
9 

670 

1 



1 
2 
3 

4 
5 
6 
7 
8 
9 

690 
1 
2 
3 
4 
5 
6 
7 



1 
8195097 
8202672 

09236 

15790 

22335 

28369 

35394 

41909 

48415 

54910 

61396 
67872 
74339 
80796 
87243 
93681 
8300109 
06528 
12937 
19337 

25728 
32109 
38480 
44843 
51196 
57540 
63374 
70199 
76516 
82822 

89120 
95409 

8401688 
07959 
14220 
20473 
26716 
32951 
39176 
45393 
51601 52221 
57800 58419 
63990 64608 
70171 70789 
76343 76960 
82507 83123 
88662 89277 
94808 95423 

8500946 01559 
07075 07687 
13195 13807 
19307 19917 
25410 26020 
31504 32113 
37590 38198 
43668 44275 
49737 50343 
55797 56403 
61849 62454 
67893 68497 
1 g 

" ' 2 



' 2 1 
9G755' 
033231 
09892 
16445 
22939^ 
29522: 
36046 
42560 
49065 
55559 
62044 
68519 
74985 
81441 
87887 
94324 
00752 
07169 
13578 
19977 
26366 
32746 
39117 
45479 
51831 
58174 
64507 
70332 
77147 
83453 

89750 
96037 
02316 



S 
8197413 
8203935 
10548 
17100 
23643 
30175 
36698 
43211 
49715 
56208 
62692 
69166 
75631 



14846 
21098 
27340 
33574 
39798 
46014 



88532 
94967 
8301394 
07811 
14218 
20616 

27005 
33384 
39754 
46114 
52465 
58807 
65140 
71463 
77778 
84083 
90379 
96666 
8402943 
09212 
15472 
21722 
27964 
34197 
40420 
46635 
52841 
59038 
65227 
71406 
77577 
83739 



4: 

98071 
04642 
11203 
17755 
24296 
30828 
37350 
43862 
50364 
56857 

63340 
69813 
76277 
82731 
89176 
95611 
02036 
08452 
14858 
21255 

27643 
34021 
40390 
46750 
53100 
59441 
65773 
72095 
78409 
84713 

.91008 
97294 
03571 



16097 
22347 



96037 
8502172 
08300 
14418 
20528 
26629 
32722 
38807 
44882 
50950 
57008 
63059 
69101 
3 



34819 
41042 
47256 

53461 
59658 
65845 
72024 
78193 
84355 
90507 
96651 
02786 
08912 

15030 
21139 
27239 
33331 
39414 
45489 
51556 
57614 
63663 
69704 
4 



5 

8198728 
8205293 
11859 
18409 
24950 
31481 
38002 
44513 
51014 
57506 

63988 
70460 
76923 
83376 
89820 
96254 

8302678 
09093 
15499 
21895 
28281 
34659 
41027 
47385 
53735 
60075 
66405 
72727 
79039 
85343 
91637 
97922 

8404198 
10465 
16723 
22971 
29211 
35442 
41664 
47877 

54081 
60277 
66463 
72641 
78810 
84970 
91122 
97264 
8503399 
09524 
15641 
21749 
27849 
33940 
40022 
46096 
52162 
58219 
64268 
70308 
5 



6 

99386 
05955 
12514 
19064 
25603 
32133 
38653 
45163 
51664 
58154 

64635 
71107 
77569 
84021 
90463 
96896 
03320 
09734 
16139 
22534 

28919 
35296 
41663 
48021 
54369 
60708 
67038 
73359 
79670 
85973 
92266 
98550 
04825 
11091 
17348 
23596 
29835 
36065 
42286 
48498 
54701 
60896 
67081 
73258 
79426 
85586 
91736 
97878 
04011 
10136 
16252 
22359 
28458 
34548 
40630 
46703 
52768 
58824 
64872 
70912 
6 



8200043 
06611 
13170 
19718 
26257 
32786 
39305 
45814 
52313 
58303 
65283 
71753 
78214 
84665 
91107 
97539 

8303962 
10375 
16778 
23173 
29558 
35933 
42299 
48656 
55003 
61341 
67670 
73990 
80301 
86602 

92895 
99178 

8405452 
11717 
17973 

' 24220 
30458 
36687 
42907 
49119 
55321 
61515 
67700 
73876 
80043 
86201 
92351 
98492 

8504624 
10748 



00700 

07268 

13825 

20372 

26910 

33438 

39956 

46464 

52963 

59451 

65931 • 

72400 

78860 

85310 

91751 

98182 

04604 

11016 

17418 

23812 

30195 

36570 

42935 

49291 

55638 

619751 

68303 

74622 

80931 

87232 



9 

8201358 
07924 
14480 
21027 
27563 
34090 
40607 
47114 
53612 
60100 

66578 
73046 
79505 
85955 
92394 
98824 
8305245 
11656 
18058 
24450 
30833 
37207 
43571 
49926 
56272 
62608 
68935 
75253 
81562 
87861 



99806 
06079 



93523 94152 



8400433 
06706 



12343 12969 

18598' 19223 

24844 1 25468 

310811 31705 



37310 
43529 
49739 
55941 
62134 
68318 
74493 



37932 
44150 
50360 
56561 
62752 
68935 
75110 



92965 
99106 
05237 
11360 
16863 17474 
22970 23580 
29068 29677 
35157 35765 
41238 41845 
47310 47917 
53374 53980 
59429 60035 
65476 66081 
71515 72118 
T 8 



80659" 81275 
87432 



93580 

99719 

8505850 

11972 

18085 
24190 
30286 
36374 
42453 
48524 
54586 
60640 
66685 
72722 
9 



dif. 

658 
7 
6 
5 
4 
3 
2 
1 


649 

8 
7 
6 
5 
4 
3 
2 
1 



639 
8 
7 
6 
5 
4 
3 
2 
1 


629 
8 
7 
7 
6 
5 
4 
3 
2 
1 


619 
8 
7 
6 
5 
5 
4 
3 
2 

1 

600 
9 
8 
7 
6 
5 
5 
4 



14 LOGARITmiS OF ^^U3IBEKS FROM 


1, TO 36,000. 


[Table i. 


Between 7200 = log — i 3-8573325, and 7800 = log "i 3-8920946. | 


tens. 


8573928 74531 8575134 75737 8576340 76943 


1 8 


9 dif.: 


720 


8577545 78148 


8578750 603 j 


1 


79955 80557 81159 81761 8-2363 3-2965 


83567 84169 


84770 2 i 


2 


35973 35575 87176 S7777 83379 339:0 


89531 90131 


SC7S2 2 { 


3 


91934 92534 93185 93785 94335 94936 


95536 96136 


96736 1 1 


4 


* 97935 93535 99185 99784 > 8600334 00933 


8601533 02132 


3602731 


518603979 04573 


8605177 05776 


06374 06973 


07571 03170 


037635599 | 


6 


09964 10562 


11160 11753 


12356 12954 


13552 14149 


14747 


8 


7 


15941 16539 


17138 17733 


13330 139-27 


195-24 20121 


20717 


7 


8 


21910 2-2507 


23103 23699 


24296 24392 


25433 26034 


26530 


7 


9 


27371 23467 


29062 -29653 


30253 30343 


31443 3-2039 


32634 


6 


|73G 


33323 34418 


35013 35603 


36202 36797 


37391 37985 


33530 


5 1 


1 39768 40362 


40956 41550 


42143 4-2737 


43331 43924 


44517 


4 1 


2 45704 46297 


46890 47433 


43076 43669 


49262 49855 


50447 


3 i 


3 51632 522-35 


52317 53409 


54001 54593 


55135 55777 


56369 


2 


4 57552 53144 


53735 59327 


59918 60509 


61100 61691 


62232 


1 


5 63464 640551 


64646 65236 


653-27 66417 


67003 67598 


63138 


1 


6 


69363 69953 


70543 71133 


71723 72317 


72907 73496 


74086 


'^i 


7 


75264 75353 


76442 77031 


775-20 73-209 


73793 79337 


79975 


589 ! 


8 


81152 81740 


8-2329 82917 


83505 84093 


84631 85-269 


85357 


8 ! 


1 9 


87032 87620 


83-207 83794 


89332 89969 


90556 91143 


91730 


7 : 


i 740 


92904 93491 


94077 94664 


95251 95337 


96423 97010 


97596 


7l 


! 1 


98733 99354 


99940/00526 
8705795' 06330 


8701112 01697 


870-2-233 02368 


8703454 


6 ! 


! 2 


87046-24 05210 


06965 07549 


03134 03719 


09304 


5 1 


3 


10473 11057 


11641 1-2-225 


1-2310 13394 


13978 14562 


15146 


4 


1 4 16313 16897' 


17430 13064 


13647 19230 


19314 20397 


20930 


3 


5 


22146 2-2723 


23311 23394 


24476 25059 


25641 26224 


26306 


2 


6 


27970 23552 


29134 29716 


30-293 30330 


31462 32043 


32625 


2 


7 


33737 34369 


34950 35531 


36112 36693 


37274 37355 


33435 


1 


8 


39597 40177 


40757 41333 


41913 42493 


43073 43658 


44233 





9 


45393 45973 


46557 47137 


47716 43296 


48375 49454 


5C034 


579 


750 


51192 51771 


52349 52923 


53507 54036 


54664 55243 


55321 


9 


1 


56973 57556 


53134 53712 


59290 59363 


60446 61023 


61601 


8 


2 


62755 63333 


63911 64438 


65065 65642 


66219 66796 


67373 


7 


3 


635-26 69103 


69630 70256 


70333 71409 


71985 7-2561 


73137 


7 


4 


74239 74365 


75441 76017 


76592 77163 


77743 78319 


73894 


6 


5 


60045 306-20 31195 817701 


82345 32919 


83494 34069 


84643 


5 


6 


85792 86367 


66941 87515 


63039 83663 


89-237 89311 


90335 


4 


7 


91532 9-2106 


92630 93-253 


93326 94400 


94973 95546 


96119 


3 


8 


97265 97333 


93411 93983 


99556/001-23 


3300701 01273 


8801845 


3 


9 


3302990 03552 


8304134 04706 


3305-278 05350 


06421 06993 


07564 


2 


760 


03707 09279 09350 10421 


10992 11563 


12134 12705 


13276 


1 


1 1 


14417 14933 15553 161-29 


16699 17-269 


17340 13410 


18930 





2 


201-20 20639 21-259 213-29 


22393 22968 


23537 24107 


24676 


569 


3 


25315 26334 26953 27522 
31502 32070 3-2639 33-207 


23090 23659 


29223 29797 


30365 


8 


4 


33775 34343 


34911 35479 


36047 


8 


5 


37132 377505 33317 33835 


39452 40019 


40586 41154 


41721 


7 


6 


42S55 43421 


43933 44555 


45122 45633 


46255 46821 


47337 


7 


7 


43520 49036 


49652 50213 


50734 51350 


51915 52481 


53047 


6 


8 


54178 54743 


55303 55374 


56439 57004 


57569 58134 


53699 


5 


9 


59323 60393 


60957 61522 


62036 62651 


63215 63779 


64343 


4 


770 


65471 66035 


66599 67163 


67726 63290 


68854 69417 


69980 


4 


1 


71107 71670 


72233 72796 


73359 73922 


74485 75043 


75610 


3 


2 


76736 77293 


77360 73423 73985 79547 


60109 60671 


81233 


2 


3 


8-2357 82918 


83430 84042 84603 85165 


65726 86287 


86848 


2 


4 


87971 83532 


89093 89653 90214 90775 


91336 91896 


92457 


1 


5 


93577 94133 


94693 95253 


95813 96373 


96933 97498 


98058 





6 


99177 99736 


8900-296 00355 


3901415 01974 


8902533 03092 


8903651 





7 


8904759 05323 


05837 06445 


07004 07563 


08121 08679 


09238 


559 


8 


10354 10912 


11470 12023 


12586 13144 


13702 14259 


14817 


8 


9 


15932 I64S9 


17047 17604 


18161 18718 


19275 19332 


20389 


7 


L— - 


1 2 


3 4^ 


5 6 


1 8 


9 






Table i.] logarithms of numbers from 


1 TO 36,000. 


^ll 


Between 7800 = log.-^ 3-8920946, and 8400 = log.^i 3-9 


212793. 1 


tens. 


1 2 


3 4 


5 6 


T 8 


9 


dif. 


780 


8921503 22059 


8922616 23173 


8923729 24285 


8924842 25398 


8925954 


556 


1 


27066 27622 


28178 28734 


29290 29846 


30401 30957 


31512 


6 


2 


32623 33178 


33733 34288 


34843 35398 


35953 36508 


37063 


5 


3 


38172 38727 


39281 39836 


40390 40944 


41498 42053 


42607 


4 


4 


43715 44268 


44822 45376 


45929 46483 


47037 47590 


48143 


4 


5 


49250 49803 


50356 50909 


51462 52015 


52568 53120 


53673 


3 


6 


54778 55330 


55883 56435 


56987 57539 


58092 58644 


59195 


2 


' 7 


60299 60851 


61403 61954 


62506 63057 


63608 64160 


64711 


2 


8 


65813 66364 


66915 67466 


68017 68568 


69118 69669 


70220 


1 


9 


71320 71871 


72421 72971 


73521 74071 


74621 75171 


75721 





790 


76821 77370 


77920 78469 


79019 79568 


80117 80667 


81216 





1 


82314 82863 


83412 83960 


84509 85058 


85606 86155 


86703 


549 


2 


87800 88348 


88897 89445 


89993 90541 


91089 91636 


92184 


8 


3 


93279 93827 


94375 94922 


95469 96017 


96564 97111 


97658 


7 


4 


98752 99299 


99846/00392 
9005310' 05856 


9000939 01486 


9002032 02579 


9003125 7 11 


5 


9004218 04764 


06402 06948 


07494 08039 


08585 


6 


6 


09676 10222 


10767 11313 


11858 12403 


12948 13493 


14038 


5 


7 


15128 15673 


16218 16762 


17307 17851 


18396 18940 


19485 


5 


8 


20573 21117 


21661 22205 


22749 23293 


23837 24381 


24924 


4 


9 


26011 26555 


27098 27641 


28185 28728 


29271 29814 


30357 


4 


BOD 


31443 31985 


32528 33071 


33613 34156 


34698 35241 


35783 


3 


1 


36867 37409 


37951 38493 


39035 39577 


40119 40661 


41202 


2 


2 


42285 42827 


43368 43909 


44450 44992 


45533 46074 


46615 


1 


3 


47696 48237 


48778 49318 


49859 50399 


50940 51480 


52020 


1 


4^ 53101 53641 


54181 54721 


55260 55800 


56340 56880 


57419 





5 


58498 59038 


59577 60116 


60655 61195 


61734 62273 


62812 


539 


6 


63889 64428 


64967 65505 


66044 66582 


67121 67659 


68197 


9 


7 


69273 69812 


70350 70887 


71425 71963 


72501 73038 


73576 


8 


8 


74651 75188 


75726 76263 


76800 77337 


77874 78411 


78948 


7 


9 


80022 80559 


81095 81632 


82169 82705 


83241 83778 


84314 


6 


810 


85386 85922 


86458 86994 


87530 88066 


88602 89137 


89673 


6 


1 


90744 91279 


91815 92350 


92885 93420 


93955 94490 


95025 


5 


' 2 


96095 96630 


97165 97699 


98234 98768 


99303 99837 


9100371 


5 


3 


9101440 01974 


9102508 03042 


9103576 04109 


9104643 05177 


05710 


4 


i 45 06778 07311 


07844 08378 


08911 09444 


09977 10510 


11043 


3 


1 5 


12109 12642 


13174 13707 


14240 14772 


15305 15837 


16369 


2 


) 6 


17434 17966 


18498 19030 


19562 20094 


20626 21157 


21689 


2 


7 


22752 23284 


23815 24346 


24878 25409 


25940 26471 


27002 


1 


i 8 


28064 28595 


29126 29656 


30187 30717 


31248 31778 


32309 


1 


9 


33369 33899 


34430 34960 


35490 36019 


36549 37079 


37609 





820 


38668 39198 


39727 40257 


40786 41315 


41844 42373 


42903 


529 


1 


43961 44489 


45018 45547 


46076 46604 


47133 47661 


48190 


9 


2 


49246 49775 


50303 50831 


51359 51887 


52415 52943 


53471 


8 


3 


54526 55054 


55581 56109 


56636 57163 


57691 58218 


58745 


7 


: 4 


59799 60326 


60853 61380 


61907 62433 


62960 63487 


64013 


7 


1 5 


65066 65592 


66118 66645 


67171 67697 


68223 68749 


69275 


6 


6 


70326 70852 


71378 71903 


72429 72954 


73479 74005 


74530 


5 


7 


75580 76105 


76630 77155 


77680 78205 


78730 79254 


79779 


5 


8 


80828 81352 


81877 82401 


82925 83449 


83973 84497 


85021 


4 


9 


86069 86593 


87117 87640 


88164 88687 


89211 89734 


90258 


4 


830 


91304 91827 


92350 92873 


93396 93919 


94442 94965 


95488 


3 


1 


96533 97055 


97578 98100 


98623 99145 


99667/00189 
9204886^05407 


9200711 


2 


2 


9201755 02277 


9202799 03321 


9203842 04364 


05929 


2 


3 


06971 07493 


08014 08535 


09056 09577 


10098 10619 


11140 


1 


4 


12181 12702 


13222 13743 


14263 14784 


15304 15824 


16345 





5 


17385 17905 


18425 18945 


19465 19984 


20504 21024 


21543 





6 


22582 23102 


23621 24140 


24659 25179 


25698 26217 


26736 


5l9 


7 


27773 28292 


28811 29330 


29848 30367 


30885 31404 


31922 


9 


8 


32958 33477 


33995 34513 


35031 35549 


36066 36584 


37102 


8 


9 


38137 38655 


39172 39690 


40207 40724 


41242 41759 


42276 


7 




1 2 


3 4 


5 6 


T 8 


9 


1 



16 


LOGARITHMS OF NUMBEKS FROM 


1 TO 36,000. 


[Table I. 1 




Between 8400 


= log -1 3-9242793, and 9000 = log.~i 3-9542425. 


tens. 


1 2 


3 4: 1 


5 6 

9245377 45394 


7 8 


9 


dif. 


840 


9243310 43827 


9244344 44860 


9246410 46927 


9247444 


517 


1 


48476 4S993 


49509 50025 


50541 51057 


51573 52039 


52605 


6 


2 


53537 54152 


54668 55184 


55699 56215 


56730 57245 


57761 


6 


3 


58791 59306 


59821 60336 


60851 61366 


61880 62395 


62910 


5 


4 


63939 64453 


64968 65482 


65997 66511 


67025 67539 


68053 


4 


5 


69081 69595 


70109 70622 


71136 71650 


72163 72677 


73190 


4 


6 


74217 74730 


75243 75757 


76270 76733 


77296 77808 


78321 


3 


7 


79347 79359 


80372 80835 


81397 81909 


82422 8-2934 


83446 


2 


8 


84471 84983 


85495 86007 


66518 87030 


87542 88054 


88565 


2 


9 


89588 90100 


90611 91123 


91634 92145 


9-2656 93167 


93676 


1 


850 


94700 95211 


95722 96233 


96743 97254 


97764 98275 


98785 


1 


1 


99806/00316 


9300826 01336 


9301847 02357 


9302866 03376 


9303386 





2 


9304906 05415 


05925 06434 


06944 07453 


07963 08472 


03931 





3 


09999 10508 


11017 11526 


12035 12544 


13053 13562 


14070 509 >| 


4 


15087 15596 


16104 16612 


17121 17629 


18137 18645 


19153^ 8 ,1 


5 


20169 20677 


21185 21692 


22200 22708 


23215 23723 


24230 
29301 


8 


6 


25245 25752 


26259 26767 


27274 27781 


28288 28795 


7 


7 


30315 30822 


31323 31835 


32341 32848 


33354 33860 


34387 


7 


8 


35379 35885 


36391 36897 


37403 37909 


38415 38920 


394-26 


6 


9 


40437 40943 


41448 41953 


42459 42964 


43469 43974 


44479 


5 


860 


45489 45994 


46499 47004 


47509 48013 


48518 49023 


49527 


5 


1 


50536 51040 


51544 52049 


52553 53057 


53561 54065 


54569 


4 


2 


55576 56080 


56584 57087 


57591 58095 


58598 59101 


59605 


4 


3 


60611 61114 


61617 62120 


62623 63126 


63629 64132 


64635 


3 


4 


65640 66143 


66645 67148 


67650 68152 


68655 69157 


69659 


3 


5 


70663 71165 


71667 72169 


72671 73172 


73674 74176 


74677 


2 


6 


75680 76182 


76683 77184 


77686 78187 


78688 79189 


■79690 


1 


7 


80692 81193 


81693 82194 


82695 83195 


83696 84196 


84697 


1 


8 


85698 86198 


86698 87198 


87698 88198 


88698 89198 


89693 





9 


90697 91197 


91697 92196 


92696 93195 


93695 94194 


94693 





870 


95692 96191 


96690 97189 


97688 98187 


98685 99184 


99683 


499 


1 


9400680 01179 


9401677 02176 


9402674 03172 


9403670 04169 


9404667 


8 


2 


05663 06161 


06659 07157 


07654 08152 


03650 09147 


09645 


8 


3 


10640 11137 


11635 12132 


12629 13126 


13623 14120 


14617 


7 


4 


15611 16108 


16605 17101 


17598 18095 


18591 19088 


19584 


7 


5 


20577 21073 


21569 22065 


22562 23053 


23553 24049 


24545 


6 


6 


25537 26032 


26523 27024 


27519 23015 


28510 29005 


29501 


6 


7 


30491 30986 


31481 31976 


32471 32966 


33461 33956 


34450 


5 


8 


35440 35934 


36429 36923 


37418 37912 


33406 38900 


39395 


5 


9 


40383 40877 


41371 41865 


42358 42852 


43346 43840 


44333 


4 


880 


45320 45814 


46307 46800 


47294 47787' 


48280 48773 


49266 


3 




50252 50745 


51238 51730 


52223 5-2716 


53208 53701 


54193 


3 


2 


55178 55671 


56163 56655 


57147 57639 


58131 58623 


59115 


2 


3 


60099 60591 


61082 61574 


62066 62557 


63049 63540 


64031 


2 


4 


65014 65505 


65996 66487 


66978 67469 


67960 68451 


68942 


1 


5 


69923 70414 


70905 71395 


71886 72376 


72866 73357 


73847 


1 


6 


74827 75317 


75807 76297 


76787 77277 


77767 78257 


78747 





' 7 


79726 80215 


80705 81194 


81684 82173 


32662 83151 


83641 


489 


1 8 


84619 85103 


85597 86035 


86574 87063 


B7552 83040 


88529 


9 


9 


89506 89995 


90483 90971 


91460 91948 


92436 92924 


93412 


8 


890 


94338 94876 


95364 95852 


96339 96827 


97315 97802 


98-290 


8 


1 


99264 99752 


9500239 00726 


9501213 01701 


9502183 02675 


9503162 


7 


2 


9504135 04622 
I 09001 09487 
i 13861 14347 


05109 05596 


06032 06569 


07055 07542 


03028 


7 


3 


09973 10459 


10946 11432 


11918 12404 


12889 


6 


4 


14832 15318 


15803 16289 


16774 17260 


17745 


6 


5 


18716 19201 


19636 20171 


20656 21141 


21626 22111 


22595 


5 


6 


23565 24049 


24534 25018 


25503 25987 


26472 26956 


27440 


5 


7 


23409 28893 


29377 29861 


30345 30823 


31312 31796 


32280 


4 


8 


33247 33731 


34214 34697 


35181 35664 
40012 40494 


36147 36631 


37114 


4 


9 


36030 38563 


39046 39529 


40977 41460 


41943 


3 


L= 


1 2 


3 4 


5 6 


T 8 


9 


ns^ 



11 Table i.] logarithms of numbers from 1 to 36,000 




17 




Between 9000 = log "i 3-9542425, and 9600 = log -^ 3-9822712. 




1^6715 


1 3 


3 4 


5 6 


7 8 


9 


dif. 


900 


9542908 43390 


9543873 44355 


9544837 45319 


9545802 46284 


9546766 


482 


1 


47730 48212 


48694 49176 


49657 50139 


50621 51102 


51584 


2 


2 


52547 53028 


53510 53991 


54472 54953 


55434 55916 


56397 


1 


3 


57358 57839 


58320 58801 


59282 5,9762 


60246 60723 


61204 


1 


4 


62165 62645 


63125 63606 


64086 64566 


65046 65526 


66006 





5 


66966 67445 


67925 68405 


68885 69364 


69844 70323 


70803 





6 


71761 72241 


72720 73199 


73678 74157 


74636 75115 


75594 


479 


7 


76552 77030 


77509 77988 


78466 78945 


79423 79902 


80360 


9 


8 


81337 81815 


82293 82771 


83249 83727 


84205 64683 


85161 


8 


9 


86117 86594 


87072 87549 


88027 88505 


88982 89459 


89937 


8 


910 


90891 91366 


91845 92322 


92800 93276 


93753 94230 


94707 


7 


1 


95660 96137 


96614 97090 


97567 98043 


98520 98996 


99472 


7 


2 


9600425 00901 


9601377 01853 


9602329 02805 


9603281 03756 


9604232 


6 


3 


05183 05659 


06135 06610 


07086 07561 


08036 08512 


08967 


6 


4 


09937 10412 


10887 11362 


11837 12312 


12767 13262 


13736 


5 


5 


14686 15160 


15635 16109 


16583 17058 


17532 18006 


16481 


5 


6 


19429 19903 


20377 20851 


21325 21799 


22272 22746 


23220 


4 


7 


24167 24640 


25114 25587 


26061 26534 


27007 27481 


27954 


4 


8 


28900 29373 


29846 30319 


30792 31264 


31737 32210 


32683 


3 


9 


33628 34100 


34573 35045 


35517 35990 


36462 36934 


37406 


2 


920 


38350 38822 


39294 39766 


40238 40710 


41161 41653 


42125 


1 


1 


43068 43539 


44011 44482 


44953 45425 


45896 46367 


46638 


2 


2 


47780 48251 


48722 49193 


49664 50135 


50605 51076 


51546 


1 


3 


52488 52958 


53428 53899 


54369 54839 


55309 55780 


56250 





4 


57190 57660 


58130 58599 


59069 59539 


60009 60478 


60948 





5 


61887 62356 


62826 63295 


63764 64233 


64703 65172 


65641 


469 


6 


66579 67048 


67517 67985 


68454 68923 


69392 69860 


70329 


9 


7 


71266 71734 


72203 72671 


73139 73607 


74076 74544 


75012 


8 


8 


75948 76416 


76884 77351 


77819 78287 


78754 79222 


79690 


8 


9 


80625 81092 


81559 82027 


82494 82961 


83428 83895 


84362 


7 


930 


85296 85763 


86230 86697 


87164 87630 


86097 88564 


89030 


7 


1 


89963 90430 


90896 91362 


91829 92295 


92761 93227 


93693 


6 


2 


94625 95091 


95557 96023 


96488 96954 


97420 97885 


98351 


6 


3 


99282 99747 


9700213 00678 


9701143 01608 


9702074 02539 


9703004 


5 


4 


9703934 04399 


04863 05328 


05793 06258 


06722 07187 


07652 


5 


5 


08581 09045 


09509 09974 


10438 10902 


11366 11830 


12294 


4 


6 


13222 13686 


14150 14614 


15078 15542 


16005 16469 


16932 


4 


7 


17859 18323 


16786 19249 


19713 20176 


20639 21102 


21565 


3 


8 


22491 22954 


23417 23880 


24343 24805 


25266 25731 


26193 


3 


9 


27118 27581 


28043 28506 


28968 29430 


29892 30354 


30816 


2 


940 


31741 32202 


32664 33126 


33588 34050 


34511 34973 


35435 


2 


1 


36358 36819 


37281 37742 


38203 38664 


39126 39587 


40048 


1 


2 


40970 41431 


41892 42353 


42814 43274 


43735 44196 


44656 


1 


3 


45577 46038 


46498 46959 


47419 47879 


46340 468G0 


49260 





4 


50180 50640 


51100 51560 


52020 52479 


52939 53399 


53856 





5 


54778 55237 


55697 56156 


56615 57075 


57534 57993 


58452 


459 


6 


59370 59829 


60288 60747 


61206 61665 


62124 62562 


63041 


9 


7 


63958 64417 


64875 65334 


65792 66251 


66709 67167 


67625 


8 


8 


68541 69000 


69458 69915 


70373 70831 


71289 71747 


72204 


8 


9 


73120 73577 


74035 74492 


74950 75407 


75864 76322 


76779 


7 


950 


77693 78150 


78607 79064 


79521 79978 


80435 80892 


81348 


7 


] 


82262 82718 


83175 83631 


84088 84544 


85001 85457 


85913 


6 


2 


86826 87282 


87738 88194 


88650 89106 


89562 90017 


90473 


6 


3 


91385 91840 
95939 96394 


92296 92751 


93207 93662 


94118 94573 


95028 


6 


4 


96649 97304 


97759 98214 


98669 99124 


99579 


5 


5 


9800488 00943 


9801398 01852 


9802307 02761 


9803216 03670 


9804125 


5 


6 


05033 05487 


05942 06396 


06850 07304 


07758 08212 


06666 


4 


7 


09573 10027 


10481 10934 


11388 11841 


12295 12748 


13202 


4 


8 


14108 14562 


15015 154.68 


15921 16374 


16827 17280 


17733 


3 


9 


18639 19092 


19544 19997 


20450 20902 


21355 21807 


22260 


3 


^ 


1 2 


3 4 


5 6 


m 8 


9 





2* 



18 


LOGARITHMS OF NUMBERS FROM 


1 TO 36,000. 


[TaWe i.l 




Between 9600 


= log -1 3-982-2712, and 10200 = log -^ 4-0086002. 


tens. 


1 2 


3 4 


5 6 


T 8 


9 


dif. 


960 


9823165 23617 


9824069 24522 


9824974 25426 


9825878 26330 


9825782 


452 


1 


27686 28138 


285S9 29041 


29493 29945 


30396 30843 


31299 


2 


2 


32202 32654 


33105 33556 


34007 34459 


34910 35361 


35812 


1 


3 


36714 37165 


37616 38066 


38517 38968 


39419 39369 


40320 


1 


4 


41221 41671 


42122 42572 


43022 43473 


43923 44373' 


44823 





5 


45723 46173 


46623 47073 


47523 47973 


48422 48872 


49322 





6 


50221 50670 


51120 51569 


52019 52468 


52917 53366 


53816 


449 


7 


54714 55163 


55612 56061 


56510 56959 


57407 57856 


58305 


9 


8 


59202 59651 


60099 60548 


60996 61445 


61893 62341 


62790 


8 


9 


63686 64134 


64582 65030 


65478 65926 


66374 66322 


67270 


8 


970 


68165 68613 


69060 69508 


69955 70403 


70850 71298 


71745 


8 


1 


72640 73087 


73534 73931 


74428 74875 


75322 75769 


76216 


7 


2 


77109 77556 


78003 78450 


78896 79343 


79789 80236 


80682 


7 


3 


81575 82021 


82467 82913 


83360 83806 


84252 84698 


85144 


6 


4 


86035 86481 


86927 87373 


87818 8S264 


83710 89155 


89601 


6 


5 


90492 90937 


91382 91828 


92273 92718 


93163 93608 


94053 


5 


6 


94943 95388 


95833 96278 


96722 97167 


97612 98057 


98501 


5 


7 


99390 99835 


9900279 00723 


9901163 01612 


9902056 02500 


9902944 


4 


8 


9903833 04277 


04721 05164 


05603 06052 


06496 06940 


07383 


4 


9 


08271 08714 


09158 09601 


10044 10488 


10931 11374 


11818 


3 


980 


12704 13147 


13590 14033 


14476 14919 


15352 15305 


16247 


3 


1 


17133 17575 


18018 18461 


18903 19345 


19738 20230 


20673 


3 


2 


21557 21999 


22441 22884 


23326 23768 


24210 24651 


25093 


2 


3 


25977 26419 


26860 27302 


27744 28185 


28627 29068 


29510 


2 


4 


30392 30834 


31275 31716 


32157 32598 


33039 33480 


33921 


1 


5 


34803 35244 


35685 36126 


36566 37007 


37448 37838 


38329 


1 


6 


39210 39650 


40090 40531 


40971 41411 


41851 42291 


42731 





7 


43612 44051 


44491 44931 


45371 45811 


46251 46690 


47130 





6 


48009 48448 


48888 49327 


49767 50206 


50645 51085 


51524 


439 


9 


52402 52841 


53280 53719 


54153 54597 


55036 55474 


55913 


9 


990 


56791 57229 


57668 58106 


58545 58983 


59422 59860 


60298 


8 


1 


61175 61613 


62051 62489 


62927 63365 


63303 64241 


64679 


8 


2 


65544 65992 


66430 66868 


67305 67743 


68130 63618 


69055 


8 


3 


69930 70367 


70804 71242 


71679 72116 


72553 72990 


73427 


7 


4 


74301 74738 


75174 75611 


76048 76485 


76921 77358 


77794 


7 


5 


78667 79104 


79540 79976 


80413 80849 


81285 81721 


82157 


6 


6 


83029 83465 


83901 84337 


84773 85209 


85645 86080 


66516 


6 


7 


87387 87823 


88258 88694 


89129 89564 


90000 90435 


90870 


5 


8 


91741 92176 


92611 93046 


93481 93916 


94350 94785 


95220 


5 


9 


96090 96524 


96959 97393 


97828 93262 


98697 99131 


99566 


4 


1000 


0000434 00869 


0001303 01737 


0002171 02605 


0003039 03473 


0003907 


4 


1 


04775 05208 


05642 06076 


06510 06943 


07377 07310 


03244 


4 


2 


09111 09544 


09977 10411 


10844 11277 


11710 12143 


12576 


3 


3 


13442 13875 


14308 14741 


15174 15607 


16039 16472 


16905 


3 


4 


17770 18202 


18635 19067 


19499 19932 


20364 20796 


21228 


2 


5 


22093 22525 


22957 23389 


23821 24253 


24635 25116 


25548 


2 


6 


26411 26843 


27275 27706 


28138 28569 


29001 29432 


29863 


1 


7 


30726 31157 


31583 32019 


32451 32882 


33313 33744 


34174 


1 


8 


35036 35467 


35898 36328 


36759 37190 


37620 38051 


38481 


1 


9 


39342 39772 


40203 40633 


41063 41493 


41924 42354 


42784 





1010 


43644 44074 


44504 44933 


45363 45793 


46223 46652 


47082 





1 


47941 48371 


48800 49229 


49659 50038 


50517 50947 


51376 


429 


2 


52234 52663 


53092 53521 


53950 54379 


54808 55237 


55666 


9 


3 


56523 56952 


57380 57809 


53233 53666 


59094 59523 


59951 


8 


4 


60808 61236 


61664 62092 


62521 62949 


63377 63805 


64233 


8 


5 


65088 65516 


65944 66372 


66799 67227 


67655 68082 


68510 


8 


6 


69365 69792 


70219 70647 


71074 71501 


71928 72355 


72782 


7 


7 


73637 74064 


74490 74917 


75344 75771 


76198 76624 


77051 


7 


8 


77904 78331 


73757 79184 


79610 80037 


80463 80889 


81316 


6 


g 


82163 82594 


83020 83446 


83872 84298 


84724 85150 


85576 


6 




1 2 


3 4 


5 6 


1 8 


9 





Table i.] logarithms of numbers from 


1 TO 36,000. 


■-19 




Between 10200 = log.~^ 4*0086002, and 10800 = log.-' 40334238. 


tens. 


1 2 


3 4 


5 6 


T 8 


9 


dif 


1020 


0086427 86853 


0087279 87704 


0088130 88556 0088981 89407 


0089832 


426 


1 


90683 91108 


91533 91959 


92384 92809 


93234 93659 


94084 


S 


2 


94934 95359 


95784 96208 


96633 97058 


97483 97907 


98332 


^ 


3 


99181 99605 


0100030 00454 


0100878 01303 0101727 02151 


0102575 


4 


4 


0103424 03S48 


04272 04696 


05120 05544 


» 05967 06391 


06815 4 II 


5 


07662 08086 


08510 08933 


09357 09780 


10204 10627 


11050 


4 


6 


11897 12320 


12743 13166 


13590 14013 


14436 14859 


15282 


3 


7 


16127 16550 


16973 17396 


17818 18241 


18664 19086 


19509 


3 


8 


20354 20776 


21198 21621 


22043 22465 


22887 23310 


23732 


2 


9 


24576 24998 


25420 25842 


26264 26685 


27107 27529 


27951 


2 


1030 


28794 29215 


29637 30059 


30480 30901 


31323 31744 


32165 


2 


1 


33008 33429 


33850 34271 


34692 35113 


35534 35955 


36376 


1 


2 


37218 37639 


38059 38480 


38901 39321 


39742 40162 


40583 


1 


3 


41424 41844 


42264 42685 


43105 43525 


43945 44365 


44785 





4 


45625 46045 


46465 46885 


47305 47725 


48144 48564 


48984' II 


5 


49823 50243 


50662 51082 


51501 51920 


52340 52759 


53178 


419 


6 


54017 54436 


54855 55274 


55693 56112 


56531 56950 


57369 


8 


7 


58206 58625 


59044 59462 


59881 60300 


60718 61137 


61555 


9 


8 


62392 62810 


63229 63647 


64065 64483 


64901 65319 


65737 


8 


9 


66573 66991 


67409 67827 


68245 68663 


69080 69498 


69916 


8 


1040 


70751 71168 


71586 72003 


72421 72838 


73256 73673 


74090 


7 


1 


74924 75342 


75759 76176 


76593 77010 


77427 77844 


78260 


7 


2 


79094 79511 


79927 80344 


80761 81177 


81594 82010 


82427 


7 


3 


83259 83676 


84092 84508 


84925 85341 


85757 86173 
89916 90332 


86589 


6 


4 


87421 87837 


88253 88669 


89084 89500 


90747' 6 If 


5 


91578 91994 


92410 92825 


93240 93656 


94071 94486 


94902 


5 


6 


95732 96147 


96562 96977 


97392 97807 


98222 98637 


99052 


5 


7 


99882/00296 
0204027^04442 


0200711 01126 


0201540 01955 


0202369 02784 


0203198 


5 


8 


04856 05270 


05684 06099 


06513 06927 


07341 


4 


9 


03169 CS583 


08997 09411 


09824 10238 


10652 11066 


11479 


4 


1050 


12307 12720 


13134 13547 


13961 14374 


14787 15201 


15614 


3 


1 


16440 16854 


17267 17680 


18093 18506 


18919 19332 


19745 


3 


2 


205/0 20983 


21396 21808 


22221 22634 


23046 23459 


23871 


3 


3 


24696 25109 


25521 25933 


26345 26758 


27170 27582 


27994 


2 


4 


28818 29230 


29642 30054 


30466 30878 


31289 31701 


32113*' 2 11 


! 5 


32936 33348 


33759 34171 


34582 34994 


35405 35817 


36228 


2 


i 6 


37050 37462 


37873 38284 


38695 39106 


39517 39928 


40339 


1 


7 


41161 41572 


41982 42393 


42804 43214 


43625 44036 


44446 


1 


8 


45267 45678 


46088 46498 


46909 47319 


47729 48139 


48549 





9 


49370 49780 


50190 50600 


51010 51419 


51829 52239 


52649 





1060 


53468 53878 


54288 54697 


55107 55516 


55926 56335 


56744 


409 


1 


57563 57972 


58382 58791 


59200 59609 


60018 60427 


60836 


9 


2 


61654 62063 


62472 62881 


63289 63698 


64107 64515 


64924 


9 


3 


65741 66150 


66558 66967 


67375 67783 


68192 68600 


69008 


8 


4 


69824 70233 


70641 71049 


71457 71865 


72273 72680 


73088 8 II 


5 


73904 74312 


74719 75127 


75535 75942 


76350 76757 


77165 


8 i 


6 


77979 78387 


78794 79201 


79609 80016 


80423 80830 


81237 


7 


7 


82051 82458 


82865 83272 


83679 84086 


84492 84899 


85306 


7 


8 


86119 86526 


86932 87339 


87745 88152 


88558 88964 


89371 


7 


9 


90183 90590 


90996 91402 


91808 92214 


92620 93026 


93432 


6 


1070 


94244 94649 


95055 95461 


95867 96272 


96678 97084 


97489 


6 


1 


98300 98706 


99111 99516 


99922/00327 
0303973^04378 


0300732 01138 


0301543 


5 


2 


0302353 02758 


0303163 03568 


04783 05188 


05592 


5 


3 


06402 06807 


07211 07616 


08020 08425 


08830 09234 


09638 


5 


4 


10447 10851 


11256 11660 


12064 12468 


12872 13277 


13681 


4 


5 


14489 14893 


15296 15700 


16104 16508 


16912 17315 


17719 


4 


6 


18526 18930 


19333 19737 


20140 20544 


20947 21350 


21754 


4 


7 


22560 22963 


23367 23770 


24173 24576 


24979 25382 


25785 


3* 


8 


26590 26993 


27396 27799 


28201 18604 


29007 29409 


29812 


3 


9 


30617 31019 


31422 31824 


32226 32629 


33031 33433 


33835 


2 




1 2 


3 4 


.5 r» 


T 8 


9 






|20 


LOGARITHMS OF NUMBERS FROM 


1 TO 36,000. 


[Table 1.1 




Between 10800 = log "i 40334238, and 11400 = log."i 40569049. .11 


tens. 


1 2 


3 4 


5 6 


7 8 


9 


dif. 


1080 


0334640 35042 


0335444 35848 


0336248 36650 


0337052 37453 


0337855 


402 


1 


38659 39060 


39462 39864 


40265 40667 


41068 41470 


41871 


2 


2 


42674 43075 


43477 43878 


44279 44680 


45081 45482 


45884 


1 


3 


466S6 47087 


47487 47883 


48289 48690 


49091 49491 


49892 


1 


4 


50693 51094 


51495 51895 


52296 52696 


53096 53497 


53897 





5 


54698 55098 


55498 55898 


56298 56698 


57098 57498 


57898 





6 


58698 59098 


59498 59898 


60297 60697 


61097 61496 


61896 





7 


62695 63094 


63494 63893 


64293 64692 


65091 65491 


65890 


399 


8 


66688 67087 


67486 67885 


68284 68683 


69082 69481 


69880 


9 


9 


70678 71076 


71475 71874 


72272 72671 


73070 73468 


73867 


9 


1090 


74663 75062 


75460 75858 


76257 76655 


77053 77451 


77849 


s 


1 


78646 79044 


79442 79839 


80237 '80635 


81033 81431 


81829 


8 


2 


82624 83022 


83419 83817 


84214 -84612 


85009 85407 


85804 


7 


3 


86599 86996 


87393 87791 


88188 88585 


88982 89379 


89776 


7 


4 


90570 90967 


91364 91761 


.92158 92554 


92951 93348 


93745 


7 


5 


94538 94934 


95331 95727 


96124 96520 


96917 97313 


97709 


6 


6 


98502 98898 


99294 99690 


0400086 00482 


0400878 01274 


0401670 


6 


7 


0402462 02858 


0403254 03650 


04045 04441 


04837 05232 


05628 


6 


8 


06419 06814 


07210 07605 


08001 08396 


08791 09187 


09582 


5 


9 


10372 10767 


11162 11557 


11952 12347 


12742 13137 


13532 


5 


1100 


14322 14716 


15111 15506 


15900 16295 


16690 17084 


17479 


5 


1 


18268 18662 


19056 19451 


19845 20239 


20633 21028 


21422 


4 


2 


22210 22604 


22998 23392 


23786 24180 


24574 24968 


25361 


4 


3 


26149 26543 


26936 27330 


27723 28117 


28510 28904 


29297 


4 


4 


30084 30477 


30871 31264 


31657 32050 


32444 32837 


33230 


3 


5 


34016 34409 


34802 35195 


35587 35980 


36373 36766 


37159 


3 


'^6 


37944 38337 


38729 39122 


39514 39907 


40299 40692 


41084 


3 


7 


41869 42261 


42653 43045 


43437 43829 


44222 44614 


45006 


2 


8 


45790 46181 


46573 46965 


47357 47749 


48140 48532 


48924 


2 


9 


49707 50099 


50490 50882 


51273 51664 


52056" 52447 


52839 


2 


1110 


53621 54012 


54403 54795 


55186 55577 


55968 56359 


56750 


1 


1 


57531 57922 


58313 58704 


59095 59485 


59876 60267 


60657 


1 


2 


61438 61829 


62219 62610 


63000 63391 


63781 64171 


64561 





3 


65342 65732 


66122 66512 


€6902 67292 


67682 68072 


68462 





4 


69242 69632 


70021 70411 


70801 71190 


71580 71970 


723591 





5 


73138 73528 


73917 74306 


74696 75085 


75474 75864 


76253 


389 


6 


77031 77420 


77809 78198 


78587 78976 


79365 79754 


80143 


9 


7 


80921 81309 


81698 82087 


82475 82864 


■83253 83641 


84030 


9 


8 


84806 85195 


85583 85972 


86360 86748 


87136 87525' 


87913 


8 


9 


88689 89077 


89465 89853 


90241 90629 


91017 91405 


91792 


8 


1120 


92568 92956 


93343 93731 


94119 94506 


94894 95281 


95669 


8 


1 


96444 96831 


97218 97606 


97993 98380 


93767 99154 


99541 


7 


2 


0500316 00703 


0501090 01477 


0501863 02250 


0502637 03024 


0503411 


7 


3 


04184 04571 


04958 05344 


05731 06117 


06504 06890 


07277 


7 


4 


08049 08436 


08822 09208 


09595 09981 


10367 10753 


11139 


6 


5 


11911 12297 


12683 13069 


13455 13841 


14227 14612 


14998 


6 


6 


15770 16155 


16541 16926 


17312 17697 


18083 13468 


18854 


6 


7 


19624 20010 


20395 20780 


21166 21551 


21036 22321 


22706 


5 


8 


23476 23861 


24246 24631 


25016 25400 


25785 26170 


26555 


5 


9 


27324 27709 


28093 28478 


28862 29247 


29631 30016 


30400 


5 


1130 


31169 31553 


31937 32321 


32706 33090 


33474 33858 


34242 


4 


1 


35010 35394 


35778 36162 


36546 36929 


37313 37697 


38081 


4 


2 


38848 39232 


39615 39999 


40382 40766 


41149 41532 


41916 


4 


3 


42682 43066 


43449 43832 


44215 44598 


44981 45365 


45748 


3 


4 


46514 46896 


47279 47662 


48045 48428 


48811 49193 


1 49576 


3 


5 


50341 50724 


51106 51489 


51871 52254 


52636 53019 


53401 


3 


6 


54166 54548 


54930 55312 


55694 56077 


56459 56841 


57223 


2 


7 


57987 58369 


58750 59132 


59514 59896 


60278 60659 


61041 


2 


8 


61804 62186 


62567 62949 


63330 63712 


64093 64475 


64856 


1 


9 


65619 66000 


66381 66762 


67143 67524 


67905 68287 


68668 


1 


'. 


1 2 


3 4 


I 5 6 


T 8 


O 






Table i.] logarithms of numbers from 


1 TO 36,000. 




21 






Between 11400 = log.-i 40569049, and 12000 = log.-^ 4079181S 






tens. 


1 Z 


3 4 


I 5 6 


1 8 


9 


dif. 




1140 


0569429 69810 


0570191 70572 


0570953 71334 


0571714 72095 


0572476 


381 




1 


73237 73618 


73998 74379 


74759 75140 


75520 75900 


76281 


1 




2 


77041 77422 


77802 78182 


78562 78942 


79322 79702 


80082 







3 


80842 81222 


81602 81982 


82362 82741 


83121 83501 


83831 


C 




4 


84640 85019 


85399 85778 


86158 86537 


86917 87296 


87676 







5 


88434 88813 


89193 89572 


89951 90330 


90709 91088 


91467 


379 




6 


92225 92604 


92983 93362 


93741 94119 


94498 94877 


95256 


9 




7 


96013 96391 


96770 97148 


97527 97905 


98284 98662 


99041 


9 




8 


99797/00175 
0603578^03956 


0600554 00932 


0601310 01688 


0602066 02444 


0602822 


8 




9 


04334 04712 


05090 05468 


05845 06223 


06601 


8 




1150 


07356 07734 


08111 08489 


08866 09244 


09621 09999 


10376 


8 




1 


11131 11508 


11885 12262 


12639 13017 


13394 13771 


14148 


7 




2 


14902 15279 


15656 16032 


16409 16786 


17163 17540 


17916 


7 




3 


18670 19046 


19423 19799 


20176 20552 


20929 21305 


21682 


7 




4 


22434 22811 


23187 23563 


23939 24316 


24692 25068 


25444 


6 




5 


26196 26572 


26948 27324 


27699 28075 


28451 28827 


29203 


6 




6 


29954 30330 


30705 31081 


31456 31832 


32207 32583 


32958 


6 




7 


33709 34084 


34460 34835 


35210 35585 


35960 36335 


36711 


5 




8 


37461 37836 


38211 38585 


38960 39335 


39710 40085 


40460 


5 




9 


41209 41584 


41958 42333 


42708 43082 


43457 43831 


44205 


5 




1160 


44954 45329 


45703 46077 


46451 46826 


47200 47574 


47948 


4 




1 


48696 49070 


49444 49818 


50192 50566 


50940 51314 


51688 


4 




2 


52435 52809 


53182 53556 


53930 54303 


54677 55050 


55424 


4 




3 


56171 56544 


56917 57291 


57664 58037 


58410 58784 


59157 


3 




4 


59903 60276 


60649 61022 


61395 61768 


62141 62514 


62886 


3 




5 


63632 64005 


64377 64750 


65123 65495 


65868 66241 


66613 


3 




6 


67358 67730 


68103 68475 


68847 69220 


69592 69964 


70336 


2 




7 


71081 71453 


71825 72197 


72569 72941 


73313 73685 


74057 


2 




8 


74800 75172 


75544 75915 


76287 76659 


77030 77402 


77774 


2 




9 


78517 78888 


79259 79631 


80002 80374 


80745 81116 


81487 


1 




1170 


82230 82601 


82972 83343 


83714 84085 


84456 84827 


85198 


1 




1 


85940 86311 


86681 87052 


87423 87794 


88164 88535 


88906 


1 




2 


89647 90017 


90388 90758 


91129 91499 


91869 92240 


92610 







3 


93350 93721 


94091 94461 


94831 95201 


95571 95941 


96311 







4 


97051 97421 


97791 98160 


98530 98900 


99270 99639 


0700009 







5 


0700748 01118 


0701487 01857 


0702226 02596 


0702965 03335 


03704 


369 




6 


04442 04812 


05181 05550 


05919 06288 


06658 07027 


07396 


9 




7 


08134 08503 


08871 09240 


09609 09978 


10347 10715 


11084 


9 




8 


11822 12190 


12559 12927 


13296 13664 


14033 14401 


14770 


9 




9 


15506 15875 


16243 16611 


16979 17348 


17716 18084 


18452 


8 




1180 


19188 19556 


19924 20292 


20660 21028 


21396 21763 


22131 


8 




1 


22867 23234 


23602 23970 


24337 24705 


25072 25440 


25807 


8 




2 


26542 26910 


27277 27644 


28011 28379 


28746 29113 


29480 


7 




3 


30215 30582 


30949 31316 


31683 32050 


32416 32783 


33150 


1 




4 


33884 34251 


34617 34984 


35351 35717 


36084 36450 


36817 


7 




5 


37550 37916 


38283 38649 


39016 39382 


39748 40114 


40481 


6 




6 


41213 41579 


41945 42311 


42677 43043 


43409 43775 


44141 


6 




7 


44873 45239 


45605 45970 


46336 46702 


47068 47433 


47799 


6 




8 


48530 48895 


49261 49626 


49992 50357 


50723 51088 


51453 


5 




9 


52184 52549 


52914 53279 


53644 54010 


54375 54740 


55105 


5 




1190 


55835 56199 


56564 56929 


57294 57659 


58024 58388 


58753 


5 




1 


59482 59847 


60211 60576 


60940 61305 


61669 62034 


62398 


4 




2 


63127 63491 


63855 64220 


64584 64948 


65312 65676 


66040 


4 




3 


66768 67132 


67496 67860 


68224 68588 


68952 69316 


69680 


4 




4 


70407 70771 


71134 71498 


71862 72225 


72589 72952 


73316 


4 




5 


74042 74406 


74769 75133 


75496 75859 


76222 76585 
79853 80216 


76949 


3 




6 


77675 78038 


78401 78764 


79127 79490 


80579 


3 




7 


81304 81667 


82030 82393 


82755 83118 


83480 83843 


84206 


3 




8 


84931 85293 


85656 86018 


86380 86743 


87105 87467 


87830 


2 




9 


88554 88916 


89278 89640 


90003 90365 


90727 91089 


91451 


2 




i— — 


1 2 


3 4 


5 6 


T 8 


9 







22 


LOGARITHMS OF NUMBERS FROM 


1 TO 36,000. 


[Table J 






Between 12000 = log. "^ 4-0791812, and 12600 = log.-'i 4-1003705 


, 




tens. 


11 SI 


3 4 1 


5 6 


T 8 


9 


dif 




i200!0792174 92536|0792898 93260S 


0793622 93983 


0794345 94707 


0795068 


362 




1 


1 95792 96153 


96515 96876 


97238 97599 


97961 98322 


98683 






2 


99406 99767 


0800128 00490 


0800851 01212k 
04461 04822 
08068 08429 


0801573 01934 


0802295 


1 




3 


0803017 03378 


0:739 04100 


05183 05543 


05904 


1 




4 


06626 06986 


07347 07707 


08789 09150 


09510 


1 




5 


10231 10591 


10952 11312 


11672 12032 


12393 12753 


13113 







6 


13S33 14193 


14553 14913 


15273 15633 


15993 16353 


16713 







7 


17432 17792 


18152 18512 


18871 19231 


19591 19950 


20310 







8 


21029 21383 


21748 22107 


22467 22826 


23185 23545 


23904 


359 




9 


24622 249S1 


25341 25700 


26059 26418 


26777 27136 


27495 


9 




1210 


28213 28571 


28930 29289 


29648 30007 


30365 30724 


31083 


9 




1 


31800 32159 


32517 32876 


33234 33593 


33951 34309 


34668 


- 8 




2 


35385 35743 


36101 36459 


36817 37176 


37534 37892 


38250 


8 




3 


38966 39324 


39682 40040 


40398 40756 


41114 41471 


41829 


8 




4 


42545 42902 


43260 43618 


43975 44333 


44690 45048 


45405 


8 




5 


46120 46478 


46835 47192 


47550 47907 


48264 48621 


48979 


7 




6 


49693 50050 


50407 50764 


51121 51478 


51835 52192 


52549 


7 




7 


53263 53619 


53976 54333 


54690 55046 


55403 55760 


56116 


7 




8 


56829 57186 


57542 57899 


58255 58612 


58968 59324 


59681 


6 




9 


60393 60750 


61106 61462 


61818 62174 


62530 62886 


63242 


6 




1220 


63954 64310 


64666 65022 


65378 65734 


66089 66445 


66801 


6 




1 


67512 67868 


68224 68579 


68935 69290 


69646 70001 


70a57 


6 




2 


71067 71423 


71778 72133 


72489 72844 


73199 73554 


73909 


5 




3 


74620 74975 


75330 75685 


76040 76395 


76750 77104 


77459 


5 




4 


78169 78524 


78878 79233 


79588 79943 


80297 80652 


81006 


5 




5 


81715 82070 


82424 82779 


83133 83488 


83842 84196 


84550 


4 




6 


85259 85613 


85967 86321 


86676 87030 


87384 87733 


88092 


4 




7 


88300 89153 


89507 89861 


90215 90569 


90923 91276 


91630 


4 




8 


92337 92691 


93045 93398 


93752 94105 


94459 94812 


95165 


4 




9 


95872 96226 


96579 96932 


97285 97639 


97992 98345 


98698 


3 




1230 


99404 99757 


0900110 00463 


0900816 01169 


0901522 01875 


0902228 


3 




1 


0902933 03286 


03639 03991 


04344 04697 


05049 05402 


05755 


3 




2 


06460 06812 


07164 07517 


07869 08222 


08574 08926 


09279 


2 




3 


09983 10335 


10687 11039 


11392 11744 


12096 12448 


12800 


2 




4 


13504 13855 


14207 14559 


14911 15263 


15614 15966 


16318 


2 




5 


17021 17373 


17724 18076 
21239 21590 
24750 25101 


18427 18779 
21941 22292 
25452 25803 


19130 19482 


19833 


2 




6 


20536 20887 


22644 22995 


23346 


1 




7 


24048 24399 


26154 26505 


26856 


1 




8 


27557 27908 


28259 28609 


28960 29311 


29661 30012 


30363 


1 




9 


31064 31414 


31764 32115 


32465 32816 


33166 33516 


33867 







1240 


34567 34917 


35267 35618 


35968 36318 


36668 37018 


37368 







1 


38063 38418 


38768 39117 


39467 39817 


40167 40517 


40866 







2 


41566 41915 


42265 42614 


42964 43313 


43663 44012 


44362 


349 




3 


45061 45410 


45759 46109 


46458 46807 


47156 47506 


47855 


9 




4 


48553 48902 


49251 49600 


49949 50298 


50647 50996 


51345 


9 




5 


52042 52391 


52740 53089 


53437 53786 


54135 54483 


54832 


9 




6 


55529 55877 


56226 56574 


56923 57271 


57620 57968 


58316 


8 




7 


59013 59361 


59709 60057 


60406 60754 


61102 61450 


61798 


8 




8 


62494 62842 


63190 63538 


63885 64233 


64581 64929 


65277 


8 




9 


65972 66320 


66667 67015 


67363 67710 


68058 68405 


68753 


8 




1250 


69448 69795 


70142 70490 


70837 71184 


71531 71879 


72226 


7 




1 


72920 73267 


73614 73962 


74309 74656 


75003 75349 


75696 


7 




2 


76390 76737 


77084 77431 


77777 78124 


78471 78817 


79164 


7 




3 


79857 80204 


80550 80897 


81243 81590 


81936 82283 


82629 


7 




4 


83322 83668 


84014 84360 


84707 85053 


85399 85745 


86091 


6 




5 


86783 87129 


87475 87821 


88167 88513 


88859 89205 


89551 


6 




6 


90242 90588 


90934 91279 


91625 91971 


92316 92662 


93007 


6 




7 


93698 94044 


94389 94735 


95080 95425 


95771 96116 


96461 


5 




8 


97152 97497 


97842 98187 


98532 98877 


99222 9.9567 


99912 


5 




9 


1000602 00947 


1001292 01637 


1001982 02327 


1002671 03016 


1003361 


5 




[__ 


1 2 


3 4- 


5 6 


7 8 


9 







Table i.] logarithms of numbers from 


1 TO 36,000. 


^55!?^™' 


23 




Between 12600 = log -^ 4-1003705, and 13200 = log.-i 4' 


1205739 




tens. 


1 2 


3 4 


5 6 


T 8 


1> 


dif. 


1260 


1004050 04395 


1004739 05084 


1005429 05773 


1006118 06462 


1006806 


345 


1 


07495 07S40 


08184 08528 


08873 09217 


09561 09905 


10249 


4 


2 


10938 11282 


11626 11970 


12314 12658 


13002 13346 


13690 


4 


3 


14377 14721 


15065 15409 


15752 16096 


16440 16784 


17127 


4 


4 


17614 18158 


18501 18845 


19188 19532 


19875 20219 


20562 


3 


5 


21249 21592 


21935 22278 


22621 22965 


23308 23651 


23994 


3 


6 


24680 25023 


25366 25709 


26052 26395 


26738 27081 


27423 


3 


7 


28109 28452 


28794 29137 


29480 29822 


30165 30507 


30850 


3 


8 


31535 31877 


32220 32562 


32905 33247 


33589 33932 


34274 


2 


9 


34958 35301 


35643 35985 


36327 36669 


37011 37353 


37695 


2 


1270 


38379 38721 


39063 39405 


39747 40089 


40430 40772 


41114 


2 


1 


41797 42139 


42480 42822 


43164 43505 


43847 44188 


44530 


2 


2 


45213 45554 


45895 46237 


46578 46919 


47260 47602 


47943 


1 


3 


48625 48966 


49307 49648 


49989 50331 


50671 51012 


51353 


1 


4 


52035 52376 


52717 53058 


53398 53739 


54080 54421 


54761 


1 


5 


55442 55783 


56124 56464 


56805 57145 


57486 57826 


58166 





6 


58847 59187 


59528 59868 


60208 60548 


60889 61229 


61569 





7 


62249 62589 


62929 63269 


63609 63949 


64289 64629 


64969 





8 


65648 65988 


66328 66668 


67007 67347 


67687 68026 


68366 





9 


69045 69385 


69724 70063 


70403 70742 


71082 71421 


71760 


339 


1280 


72439 72778 


73117 73457 


73796 74135 


74474 74813 


75152 


9 


1 1 


75830 76169 


76508 76847 


77186 77525 


77864 78203 


78541 


9 


2 


79219 79558 


79896 80235 


80574 80912 


81251 81590 


81928 


9 


3 


82605 82944 


83282 83620 


83959 84297 


84635 84974 


85312 


8 


4 


85988 86327 


86665 87003 


87341 87679 


88017 88355 


88693 


8 


5 


89369 89707 


90045 90883 


90721 91059 


91396 91734 


92072 


8 


6 


92747 93085 


93423 93760 


94098 94435 


94773 95111 


95488 


8 


7 


96123 96460 


96798 97135 


97472 97810 


98147 98484 


98821 


7 


8 


99496 99833 


1100170 00507 


1100844 01181 


1101518 01855 


1102192 


7 


9 


1102866 03203 


03540 03877 


04213 04550 


04887 05224 


05560 


7 


1290 


06234 06570 


06907 07244 


07580 07917 


08253 08590 


08926 


7 


1 


09599 99935 


10272 10608 


10944 11280 


11617 11953 


12289 


6 


2 


12961 13297 


13633 13969 


14306 14642 


14977 15313 


15649 


6 


3 


16321 16657 


16993 17329 


17664 18000 


18336 18671 


19007 


6 


4 


19678 20014 


20350 20685 


21021 21356 


21691 22027 


22362 


6 


1 5 


23033 23368 


23704 24039 


24374 24709 


25045 25380 


25715 


5 


1 6 


26385 26720 


27055 27390 


27725 28060 


28395 28730 


29065 


5 


7 


29735 30069 


30404 30739 


31074 31408 


31743 32078 


32412 


5 


8 


33081 33416 


33751 34085 


34420 34754 


35088 35423 


35757 


4 


9 


36426 36760 


37094 37429 


37763 38097 


38431 38765 


39099 


4! 


1300 


39768 40102 


50436 40770 


41104 41437 


41771 42105 


42439 


4 

4 [ 


1 


43107 43441 


43774 44108 


44442 44775 


45109 45443 


45776 


2 


46443 46777 


47110 47444 


47777 84111 


48444 48777 


49111 


3l 


3 


49777 50111 


50444 50777 


51110 51444 


51777 52110 


52443 


3| 


4 


53109 53442 


53775 54108 


54441 54774 


55107 55439 


55772 


3 


5 


56438 56771 


57103 57436 


57769 58101 


58434 58767 


59099 


3 


6 


59764 60097 


60429 60762 


61094 61427 


61759 62091 


62424 


2 


7 


63088 63420 


63753 64085 


64417 64749 


65081 65413 


65745 


2 


8 


66409 66741 


67073 67405 


67737 68069 


68401 68733 


69065 


2 


9 


69728 70060 


70392 70723 


71055 71387 


71718 72050 


72381 


2 


1310 


73044 73376 


73707 74039 


74370 74702 


75033 75364 


75696 


1 


1 


76358 76689 


77021 77352 


77683 78014 


78345 78676 


79007 


1 


9 


79669 80000 


80331 80662 


80993 81324 


81655 81986 


82316 


1 


3 


82978 83309 


83639 83970 


84301 84631 


84962 85293 


85623 


1 


4 


86284 86615 


86945 87276 


87606 87936 


88267 88597 


88927 





5 


89583 89918 


90248 90578 
93549 93879 


90909 91239 


91561 91899 


92229 





6 


92889 93219 


94209 94539 


94868 95198 


95528 





7 


96187 96517 


96847 97177 


97506 97836 


98165 98495 


9S325 





8 


99484 99813 


1200143 00472 


1200R01 01131 
04094 04423 
5 6 


1201460 01789 


1202119 
05410 
9 


329 


9 


1202777 03106 


03436 03765 


04752 05081 


9 





1 2 


3 4 


7 8 





24 


tOGAPJTHMS OF NUMBERS FROM 


1 TO 36,000. 


[Table I. 


: 




Between 13200 = log "i 4-1205739, and 13800 = log -^ 4-1398791. 




tens. 


1 2 


3 4 


5 6 

1207334 07713 


T 8 


9 


dif 




1320 


1206068 06397 


1206726 07055 


1208042 08371 


1203699 


329 




1 


09357 09686 


10014 10343 


10672 11000 


11329 11657 


11936 


9 




2 


12643 12972 


13300 13628 


13957 14235 


14614 14942 


15270 


8 




3 


15927 16255 


16583 16911 


17239 17563 


17896 16224 


18552 


8 




4 


19208 19536 


19864 20192 


20520 20348 


21175 21503 


21831 


8 




5 


22487 22814 


23142 23470 


23797 24125 


24453 24780 


25108 


8 




6 


25763 26090 


26418 26745 


27073 27400 


27727 23055 


28382 


7 




7 


29036 29364 


29691 30018 


30345 30672 


31000 31327 


31654 


7 




8 


32308 32635 


32962 33239 


33616 33942 


34269 34596 


34923 


7 




9 


35577 35903 


36230 36557 


36883 37210 


37537 37663 


36190 


7 




1330 


33843 39169 


39496 39322 


40149 40475 


40302 41128 


41454 


6 




1 


42107 42433 


42759 43036 


43412 43733 


44064 44390 


44716 


6 




2 


45368 45694 


46020 46346 


46672 46993 


47324 47650 


47976 


6 




3 


48627 43953 


49279 49605 


49930 50256 


50582 50907 


51233 


6 




4 


51834 52209 


52535 52360 


53136 53511 


53337 54162 


54487 


5 




5 


55138 55463 


55763 56114 


56439 56764 


57069 57414 


57739 


5 




6 


58390 58715 


59040 59365 


59690 60015 


60339 60664 


60989 


5 




7 


61639 61964 


62233 62613 


62933 63263 


63567 63912 


64237 


5 




8 


648S6 65210 


65535 65359 


66134 66508 


66333 67157 


67481 


4 




9 


68130 6S454 


63779 69103 


69427 69751 


70076 70400 


70724 


4 




1340 


71372 71696 


72020 72344 


72663 72992 


73316 73640 


73964 


4 




1 


74612 74935 


75259 75533 


75907 76230 


76554 76378 


77202 


4 




2 


77849 78172 


78496 78319 


79143 79466 


79790 60113 


80437 


4 




3 


81033 81407 


81730 82053 


82377 82700 


83023 83346 


63670 


3 




4 


84316 84639 


64962 85285 


65603 85931 


86254 86577 


86900 


3 




5 


87546 87369 


63191 63514 


88337 69160 


89483 69605 


90128 


3 




6 


90773 91096 


91413 91741 


92064 92386 


92709 93031 


93354 


3 




7 


93993 94321 


94643 94965 


95263 95610 


95932 96255 


96577 


2 




8 


97221 97543 


97665 98187 


98510 95332 


99154 99476 


99796 


2 




9 


1300441 00763 


1301035 01407 


1301729 02051 


1302372 02694 


1303016 


2 




1350 


03659 03981 


04303 04624 


04946 05267 


05569 05911 


06232 


2 




1 


06375 07196 


07518 07839 


08161 03432 


06803 09124 


09446 


1 




2 


10033 10409 


10730 11052 


11373 11694 


12015 12336 


12657 


1 




3 


13299 13620 


13941 14262 


14533 14903 


15224 15545 


15866 


1 




4 


16507 16323 


17149 17469 


17790 18111 


18431 18752 


19072 


1 




5 


19713 20034 


20354 20675 


20995 21316 


21636 21956 


22277 







6 


22917 23237 


23553 23373 


24193 24518 


24333 25158 


25478 







7 


26119 26439 


26758 27073 


27393 27718 


28033 28356 


28678 







8 


29317 29637 


29957 30277 


30596 30916 


31236 31555 


31875 







9 


32514 32334 


33153 33473 


33792 34112 


34431 34750 


35070 


319 




1360 


35703 36023 


35347 36666 


36935 37305 


37624 37943 


38252 


9 




1 


33900 39219 


39538 39857 


40176 40495 


40814 41133 


41452 


9 




2 


42090 42409 


42728 43046 


43365 43684 


44003 44321 


44640 


9 




3 


45277 45596 


45914 46233 


46551 46870 


47183 47507 


47825 


8 




4 


43462 48730 


49099 49417 


49735 50054 


50372 50690 


51008 


8 




5 


51645 51963 


52231 52599 


52917 53235 


53553 53371 


54169 


8 




6 


54325 55143 


55461 55779 


56096 56414 


56732 57050 


57367 


8 




7 


53003 53320 


53633 53956 


59273 59591 


59903 60226 


60543 


8 




8 


61173 61496 


61813 62131 


62448 62765 


63033 63400 


63717 


7 




9 


64352 64669 


64936 65303 


65620 65937 


66255 66572 


66889 


7 




1370 


67523 67340 


68157 68473 


68790 69107 


69424 69741 


70058 


7 




1 


70691 71003 


71325 71641 


71958 72275 


72591 72908 


73225 


7 




2 


73858 74174 


74491 74807 


75124 75440 


75756 76073 


76389 


6 




3 


77022 77338 


77654 77970 


78287 78603 


78919 79235 


79551 


6 




4 


60183 80499 


80815 81131 


81447 81763 


82079 82395 


82711 


6 




5 


83343 83659 


83974 84290 


64606 84922 


85237 85553 


85869 


6 




6 


86500 86316 


87131 87447 


87762 86078 


88393 88709 


89024 


5 




7 


89655 89970 


90285 90601 


90916 91231 


91547 91862 


92177 


5 




8 


92807 93122 


93438 93753 


94063 94333 


94698 95013 


95328 


5 




9 


95958 96272 


96587 96902 


97217 97532 


97847 98161 


98476 


5 






1 2 


3 4 


5 6 


7 8 


9 







Table i.] logarithms of 


NUMBERS FROM 1 TO 36,000 


251 




Between 13800 


= log -1 4- 1398791, and 14400 =: log "i 4' 1583625. 




tens. 


1 2 


3 4 


1 5 6 


T 8 


9 


dif. 




13S0 


1399106 99420 


1399735/00050 
1402880^03195 


1400364 00679 


1400993 01308 


1401622 


315 




1 


1402251 02566 


03509 03823 


04138 04452 


04766 






2 


05395 05709 


06023 06337 


06651 06966 


07280 07594 


07908 






3 


08536 08350 


09164 09478 


09792 10106 


10419 10733 


11047 






4 


11675 11988 


12302 12616 


12930 13243 


13557 13371 


14184 






5 


14811 15125 


15438 15752 


16065 16379 


16692 17006 


17319 






6 


17946 18259 


18572 18885 


19199 19512 


19825 20138 


20451 


3 




7 


21078 21391 


21704 22017 


22330 22643 


22956 23269 


23582 


3 




8 


24208 24520 


24833 25146 


25459 25772 


26084 26397 


26710 


3 




9 


27335 27648 


27960 28273 


28586 28898 


29211 29523 


29836 


3 




1390 


30460 30773 


31085 31398 


31710 32022 


32335 32647 


32959 


2 




1 


33584 33896 


34208 34520 


34832 35144 


35456 35768 


36080 


2 




2 


36704 37016 


37328 37640 


37952 38264 


38576 38888 


39199 


2 




3 


39823 40135 


40446 40758 


41070 41381 


41693 42005 


42316 


2 




4 


42939 43251 


43562 43874 


44185 44497 


44808 45119 


45431 


1 




5 


46053 46365 


46676 46987 


47298 47610 


47921 48232 


48543 


1 




6 


49165 49476 


49787 50093 


50409 50720 


51031 51342 


51653 


1 




7 


52275 52586 


52897 53207 


53513 53329 


54140 54450 


54761 


1 




8 


55382 55693 


56004 56314 


56625 56935 


57246 57556 


57867 







9 


58488 58798 


59108 59419 


59729 60039 


60350 60660 


60970 







1400 


61591 61901 


62211 62521 


62831 63141 


63451 63761 


64071 







1 


64691 65001 


65311 65621 


65931 66241 


66551 66861 


67170 







2 


67790 68100 


68409 68719 


69029 69338 


69648 69958 


70267 







3 


70886 71196 


71505 71815 


72124 72434 


72743 73052 


73362 


309 




4 


73980 74290 


74599 74908 


75217 75527 


75836 76145 


76454 


9 




5 


77072 77381 


77690 77999 


■ 78308 78617 


78826 79235 


79544 


9 




6 


80162 80471 


80730 81039 


81397 81706 


82015 32324 


82632 


9 




7 


83250 83558 


83867 34175 


84484 84793 


85101 85410 


85713 


9 




8 


85335 86643 


86952 87260 


87569 87877 


88185 88493 


38802 


8 




9 


89418 89726 


90035 90343 


90651 90959 


91267 91575 


91883 


8 




1410 


92499 92307 


93115 93423 


93731 94039 


94347 94655 


94962 


8 




1 


95578 95886 


96193 96501 


96809 97116 


97424 97732 


98039 


8 




2 


98655 98962 


99270 99577 


99835/00192 
1502958^03265 


1500499 00807 


1501114 


7 




3 


1501729 02036 


1502344 02651 


03573 03830 


04187 


7 




4 


04801 05103 


05415 05722 


06030 06337 


06644 06951 


07257 


7 




5 


07371 03173 


08485 08792 


09099 09406 


09712 10019 


10326 


7 




6 


10939 11246 


11553 11359 


12166 12472 


12779 13085 


13392 


7 




7 


14005 14311 


14618 14924 


15231 15537 


15843 16150 


16456 


6 




8 


17069 17375 


17631 17937 


18293 18600 


13906 19212 


19518 


6 




9 


20130 20436 


20742 21048 


21354 21660 


21966 22272 


22578 


6 




1420 


23189 23495 


23801 24107 


24412 24718 


25024 25329 


25635 


6 




1 


26246 26552 


26358 27163 


27469 27774 


28080 28385 


28691 


6 




2 


29301 29607 


29912 30217 


30523 30328 


31133 31439 


31744 


5 




3 


32354 32659 


32964 33270 


33575 33880 


34185 34490 


34795 


5 




4 


35405 35710 


36015 36320 


36625 36929 


37234 37539 


37844 


5 




5 


38453 38753 


39063 39368 


39672 39977 


40281 40536 


40891 


5 




6 


41500 41804 


42109 42413 


42713 43022 


43327 43631 


43935 


5 




7 


44544 44348 


45153 45457 


45761 46065 


46370 46674 


46973 


4 




8 


47586 47890 


48194 48493 


48802 49106 


49410 49714 


50018 


4 




9 


50626 50930 


51234 51533 


51842 52145 


52449 52753 


53057 


4 




1430 


53664 53968 


54271 54575 


54879 55182 


55486 55789 


56093 


4 




1 


56700 57003 


57307 57610 


57914 58217 


58520 58824 


59127 


3 




2 


59733 60037 


60340 60643 


60946 61249 


61553 61856 


62159 


3 




3 


62765 63068 


63371 63674 


63977 64280 


64583 64886 


65189 


3 




4 


65794 66097 


66400 66703 


67006 67308 


67611 67914 


68216 


3 




5 


68822 69124 


69427 69729 


70032 70334 


70637 70939 


71242 


3 




6 


71847 72149 


72452 72754 


73056 73359 


73661 73963 


74265 


2 




7 


74870 75172 


75474 75776 


76079 76381 


76683 76985 


77287 


2 




8 


77891 78193 


78495 78797 


79099 79401 


79702 80004 


80306 


2 ' 




L 


80910 81212 


81513 81815 


82117 82418 


82720 83022 


83323 


2 ! 




1 2 


3 4 


5 6 


T 8 


9 


iisJ 





2Q 


LOGARITHMS OF NUMBERS FROM 1 TO 36,000. [Table I. 


Between 14400 = logn 4-1583625, and 15000 = log.-i 4-1760913. 


tens. 


1 


2 


3 4 


5 


6 , T 


8 


9 


dif. 


1440 


1583927 


4228 


1584530 4831 


1585133 


5434 


1585736 


6037 


1586338 


301 


1 


6941 


7243 


7544 7845 


8146 


8443 


8749 


9050 


9351 




2 


9954 


/0255 
^3265 


1590556 0357 


1591158 


1459 


1591760 


2061 


1592362 




3 


1592964 


3566 3867 


4168 


4469 


4770 


5070 


5371 




4 


5973 


6273 


6574 6375 


7175 


7476 


7777 


8077 


8378 




5 


8979 


9230 


9530 9381 


1600181 


0481 


1600762 


1032 


1601383 




6 


I6019S3 


2284 


1602534 2834 


3184 


3435 


3735 


4035 


4335 





7 


4985 


5286 


5586 5886 


6186 


6436 


6786 


7036 


7336 





8 


7986 


8285 


8585 8835 


9185 


9435 


9785 


y0034 

'3031 


1610334 





9 


1610984 


1283 


1611533 1883 


1612182 


2482 


1612781 


3380 





1450 


3980 


4279 


4578 4878 


5177 


5477 


5776 


6075 


6375 





^ 


6973 


7273 


7572 7871 


8170 


8470 


8769 


9068 


9367 


299 


2 


9965 


/0264 
3254 


1620563 0362 


1621161 


1460 


1621759 


2058 


1622357 


9 


3 


1622955 ' 


3553 3352 


4150 


4449 


4743 


5047 


5345 


9 


4 


6943 


6241' 6540 6339' 7137 


7436 


7734 


8033 


8331" 9 


5 


8928 


9227 


9525 9324 


1630122 


0420 


1630719 


1017 


1631315 


8 


6 


1631912 


2210 


1632503 2307 


3105 


3403 


3701 


3999 


. 4297 


8 


7 


4894 


5192 


5490 5733 


6086 


6334 


6632 


6979 


7277 


8 


8 


7873 


8171 


8469 8767 


9064 


9362 


9660 


9958 


1640255 


8 


9 


1640851 


1148 


1641446 1743 


1642041 


2339 


1642836 


2934 


3231 


8 


1460 


3826 


4123 


4421 4718 


5016 


5313 


5610 


5908 


6205 


7 


1 


6799 


7097 


7394 7691 


7933 


8235 8582 
1256 1651553 
4224 4521 


8880 


9177 


7 


2 


9771 


^0068 
3037 


1650365 0662 


1650959 


1850 


1652146 


7 


3 


1652740 ' 


3334 3631 


3927 


4817 


5114 


7 


4 


5707 


6004 


6301 6597 6894 


7190* 7437 


7783 


8080 


7 


5 


8673 


8969 


9265 9562 


9858 


,0155 


1660451 


0747 


1661043 


6 


6 


1661636 


1932 


1662223 2525 


1662321/31171 


3413 


3709 


4005 


6 


7 


4597 


4893 


5189 5485 


5781 


6077 


6373 


6669 


6965 


6 


6 


7556 


7852 


8148 8444 


8740 


9035 


9331 


9627 


9922 


6 


9 


1670514 


0309 


1671105 1400 


1671696 


1991 


1672287 


2582 


1672878 


6 


1470 


3469 


3764 


4060 4355 


4650 


4946 


5241 


5536 


5831 


5 


1 


6422 


6717 


7012 7308 


7603 


7898 


8193 


8483 


8783 


5 


2 


9373 


9668 


9963 /0253 
1682912^3207 


1680553 


0343 


1681143 


1433 


1681733 


5 


3 


1682322 


2617 


3501 


3796 


4091 


4336 


4680 


5 


4 5269 


5564 5859 6153". 


6443 


6742 


7037 


7331 7626 


5 


5 


8215 


8509 


8803 9098 


9392 


9636 


9981 


.02751 


1690589 


4 


6 


1691158 


1452 


1691746 2040 


1692335 


2829 


1692923/32171 


3511 


4 


7 


4099 


4393 


4687 4981 


5275 


5569 


5363 


6157 


6450 


4 


8 


7038 


7332 


7626 7920 


8213 


8507 


8801 


9094 


9383 


4 


9 


9975 


,0269 


1700563 0856 


1701150 


1443 


1701737 


2030 


1702324 


4 


1480 


1702911^32041 


3497 3791 


4084 


4377 


4671 


4964 


5257 


3 


1 


5844 


6137 


6430 6723 


7017 


7310 


7603 


7896 


8189 


3 


2 


8775 


9068 


9361 9654 


9947, 


0240 


1710533 


0326 


1711119 


3 


3 


1711704 


1997 


1712290 2583 


1712376/ 


3163 


3461 


3754 


4046 


3 


4 


4632 


4924 


5217 5509 


5802 


6095 


6387 


6630 


6972 


3 


5 


7557 


7849 


8142 8434 


8727 


9019 


9311 


9604 


9898 


2 


6 


1720480 


0773 


1721065 1357 


1721649 


1941 


1722233 


2526 


1722818 


2 


7 


3402 


3694 


3986 4278 


4570 


4362 


5154 


5446 


5737 


2 


8 


6321 


6613 


6905 7197 


7488 


7780 


8072 


8384 


8855 


2 


9 


9239 


9530 


9822 0113 


1730405 


0697 


1730988 


1280 


1731571 


1 


1490 


1732154 


2446 


1732737/3028 


3320 


3611 


3903 


4194 


4485 


1 


•1 


5068 


5359 


5650 5941 


6233 


6524 


6815 


7106 


7397 


1 


2 


7979 


8270 


8561 8852 


9143 


9434 


9725 


.0016 


1740307 


1 


3 


1740889 


1180 


1741471 1761 


1742052 


2343 


1742634/29251 


3215 


1 


4 cj/yy 


4087 


4378 4669 


4959 


5250 


5540 


5831" 


6121 





5 """ 


b7U2 


6993 


7283 7574 


7864 


8155 


8445 


8735 


9026 





6 


9606 


9897 


1750187 0477 


1750767 


1057 


1751348 


1638 


1751928 





7 


1752508 


2798 


3083 3378 


3668 


3958 


4248 


4538 


4828 





8 


5408 


5698 


5988 6278 


6567 


6857 


7147 


7437 


7727 





9 


8306 


8596 


8885 9175 


9465 


9754 


1760044 


0333 


1760623 





1 1 


'5 S a * s 5 


6 


7 


8 


9 Jl 



Table i.] 


LOGARITHMS OF NUMBERS FROM 1 TO 36,000. 27 | 


Between 15000 = log.-^ 4- 1760913, and 15600 = log "i 4-1931246. 1 


^'tens. 1 


2 


I 3 4 


5 6 


7 8 


9 


dif. 


1500 1761202 


1492 


1761781 2071 


1762360 2649 


1762939 3228 


1763518 


289 


1 4096 


4386 


4875 4964 


5253 5543 


5832 6121 


6410 


9 


2 6983 


7278 


7567 7856 


8145 8434 


8723 9012 


9301 


9 


3 9879 


/0168 


1770457 0745 


1771034 1323 


1771612 1901 


1772190 


9 


4 1772767 


'3056' 3345 3633' 3922 4211 


4499 4788 5076» 9 || 


5 


5654 


5942 


6231 6519 


6808 7096 


7385 7673 


7961 


8 


6 


8538 


8826 


9115 9403 


9691 9980 


1780268 0556 


1780844 


8 


7 


17S1421 


1709 


1731997 2285 


1782573 2861 


3149 3437 


3725 


8 


8 


4301 


4539 


4877 5165 


5453 5741 


6029 6317 


6605 


8 


9 


7180 


7468 


7756 8043 


8331 8619 


8907 9194 


9482 


8 


1510 


1790057 


0345 


1790632 0920 


1791207 1495 


1791782 2070 


1792357 


8 


1 


2932 


3219 


3507 3794 


4082 4369 


4656 4943 


5231 




2 


5805 


6092 


6380 6667 


6954 7241 


7528 7815 


8102 




3 


8676 


8963 


9250 9537 


9824 /Olll 


1800398 0685 


1800972 




4 '1801546 


1832'1802119 2406'1802693' 2980 


3266 3553' 3840 




5 


4413 


4700 


4986 52731 5559 5846 


6133 6419 


6706 




6 


7278 


7565 


7851 8138 


8424 8711 


8997 9283 


9570 




7 


1810142 


0428 


1810715 1001 


1811287 1573 


1811859 2145 


1812432 


6 


8 


3004 


3290 


3576 3862 


4148 4434 


4720 5006 


5292 


6 


9 


5864 


6150 


6435 6721 


7007 7293 


7579 7864 


6150 


6 


1520 


8722 


9007 


9293 9579 


9864 /0150 
1822720 ' 3005 


1820435 0721 


1821007 


6 


1 


1821578 


1863 


1822149 2434 


3290 3576 


3861 


6 


2 


4432 


4717 


5002 5288 


5573 5858 


6143 6429 


6714 


5 


3 


7284 


7569 


7854 8140 


8425 8710 


8995 9280 


9565 


5 


i 4*1830135 


0420*1830704 0989"1831274 1559'1831844 2129'1832414 


5 


5 


2983 


3268 


3553 3837 


4122 4407 


4691 4976 


5261 


5 


6 


5830 


6114 


6399 6664 


6968 7253 


7537 7822 


8106 


5 


7 


8675 


8959 


9244 9528 


9812 / 0096 
1842654 ' 2939 


1840381 0665 


1840949 


4 


8 


1841518 


1802 


1842086 2370 


3223 3507 


3791 


4 


9 


4359 


4643 


4927 5211 


5495 5779 


6063 6347 


6630 


4 


1530 


7198 


7482 


7766 8050 


8333 8617 


8901 9185 


9468 


4 


1 


1850036 


0319 


1850603 0886 


1851170 1454 


1851737 2021 


1852304 


4 


2 


2871 


3155 


3438 3721 


4005 4288 


4572 4855 


5138 


3 


3 


5705 


5988 


6271 6555 


6838 7121 


7404 7687 


7970 


3 


4 8537 


8820 


9103 9386 9669 9952=1860235 0518 


1860801' 3 II 


5 


1861367 


1650 


1861932 2215 


1862498 2781 


3064 3347 


3629 


3 


6 


4195 


4478 


4760 5043 


5326 5608 


5891 6174 


6456 


3 


7 


7021 


7304 


7586 7869 


8151 8434 


8716 8999 


9281 


2 


8 


9846/ 
1872668' 


0128 


1870410 0693 


1870975 1257 


1871540 1822 


1872104 


2 


1 9 


2951 


3233 3515 


3797 4079 


4361 4643 


4925 


2 


11540 


5489 


5771 


6053 6335 


6617 6899 


7181 7463 


7745 


2 


1 


8308 


8590 


8872 9154 


9435 6717 


999.9 / 0280 1880562 
1882815 ' 3096 3378 


2 


2 


1881125 


1407 


1881689 1970 


1882252 2533 


2 


3 


3941 


4222 


4504 4785 


5066 5348 


5629 5910 6192 




4 


6754 


7035 


7317 7598 


7879 8160 


8441 8723 9004 




5 


9566 


9847 


1890128 0409 


1890690 0971 


1891252 1533 


1891814 




3 


1892376 


2657 


2938 3218 


3499 3780 


4061 4342 


4622 




7 


5184 


5465 


5745 6026 


6307 6587 


6868 7148 


7429 


1 


8 


7990 


8271 


8551 8832 


9112 9393 


9673 9953 


1900234 





9 


1900795 


1075 


1901355 1636 


1901916 2196 


1902476 2757 


3037 





1550 


3597 


3877 


4157 4438 


4718 4998 


5278 5558 


5838 





1 


6398 


6678 


6958 7238 


• 7518 7798 


8078 8357 


8637 





2 


9197 


9477 


9757 / 0036 
1912553' 2833 


1910316 0596 


1910876 1155 


1911435 





3 


1911994 


2274 


3113 3392 


3672 3951 


4231 





4 


4790 


5060 


5348 5628 


5907 6187 


6466 6745 


7025 


279 


5 


7583 


7862 


8142 8421 8700 8979 f 9259 9538 


9817 


9 


6 


1920375 


0654 


1920933 1212 


1921491 1770 


1922049 2328 


1922607 


9 


7 


3165 


3444 


3723 4002 


4281 4559 


4838 5117 


5396 


9 


8 


5953 


6232 


6511 6789 


7068 7347 


7625 7904 


8183 


9 


9 


8740 


9018 


9297 9575 


9854 / 0132 
5 ' 6 


1930411 0689 


1930968 


9 




1 


2 


3 4 


T 8 


9 


=J 





28 LOGARITHMS OF NUMBERS FROM 1 TO 36,000. [Table I. [| 




Between 


15600 = log.-i 4- 1931246, and 16200 = log 


^ -14-2095150. || 




tens. 


1 


2 


3 ^ 1 


5 


6 


t 


8 


9 


dif. 




1560 


1931524 


1803 


1932081 23591 


1932638 


2916 


1933194 


3473 


1933751 


278 




1 


4307 


4585 


4864 51421 


5420 


5698 


5976 


6254 


6532 


8 




2 


7088 


7366 


7644 7922 


8200 


8478 


8756 


9034 


9312 


8 




3 


9868/01451 


1940423 0701 


1940979 


1257 


1941534 


1812 


1942090 


8 




4 


1942645 ' 


2923" 


3200 3478 


3756 


4033 


4311 


4588 4866 » 


8 




5 


5421 


5698 


5976 6253 


6531 


6808 


7086 


7363 


7640 


7 




6 


8195 


8472 


8749 9027 


9304 


9531 


9858/ 
1952630 ' 


0136 


1950413 


7 




7 


1950967 


1244 


1951521 1798 


1952075 


2353 


2907 


3184 


7 




8 


3733 


4014 


4291 4568 


4845 


5122 


5399 


5676 


5953 


7 




9 


6506 


6783 


7060 7336 


7613 


7890 


8167 


8443 


8720 


7 




1570 


9273 


9550 


9826/0103 
1962591 ' 2867 


1960379 


0656 


1960932 


1209 


1961485 


7 




1 


1962038 


2315 


3144 


3420 


3697 


3973 


4249 


6 




2 


4802 


5078 


5354 5630 


5907 


6183 


6459 


6735 


7011 


6 




3 


7563 


7839 


8115 8391 


8667 


8943 


9219 


9495 


9771 


6 




4 


1970323 


0599 


1970875 1151 1971427 


1702 1971978 


2254 


1972530 


6 




5 


3081 


3357 


3633 3908 


4184 


4460 


4735 


5011 


5287 


6 




6 


5838 


6113 


6389 6664 


6940 


7215 


7491 


7766 


8042 


6 




7 


8592 


8868 


9143 9418 


9694 


9969 


1980244 


0520 


1980795 


5 




8 


1981345 


1620 


1981896 2171 


1982446 


2721 


2996 


3271 


3546 


5 




9 


4096 


4371 


4646 4921 


5196 


5471 


5746 


6021 


6296 


5 




1580 


6846 


7121 


7395 7670 


7945 


8220 


8495 


8769 


9044 


5 




1 


9593 


9868 


1990143 0417 


1990692 


0967 


1991241 


1516 


1991790 


5 




2 


1992339 


2614 


2888 3163 


3437 


3712 


3986 


4260 


4535 


4 




3 


5083 


5358 


5632 5906 


6181 


6455 


6729 


7003 


7278 


4 




4 


7826 


8100 


8374 8648 


8922 


919? 


9471 


9745 


2000019 


4 




5 


2000567 


0841 


2001115 1389 


2001662 


1936 


2002210 


2484 


2758 


4 




6 


3306 


3579 


3853 4127 


4401 


4674 


• 4948 


5222 


5496 


4 




7 


6043 


6317 


6590 6864 


7137 


7411 


7684 


7958 


8231 


4 




8 


8778 


9052 


9325 9599 


9872 


,0146 


2010419 


0692 


2010966 


3 




9 


2011512 


1786 


2012059 2332 


2012605/2879 


3152 


3425 


3698 


3 




1590 


4244 


4517 


4791 5064 


5337 


5610 


5883 


6156 


6429 


3 




1 


6975 


7248 


7521 7794 


8066 


8339 


8612 


8885 


9158 


3 




2 


9703 


9976 


2020249 0522 


2020794 


1067 


2021340 


1612 


2021885 


3 




3 


2022430 


2703 


2976 3248 


3521 


3793 


4066 


4338 


4611 


3 




4 


5156 


5428 


5700 5973 


6245 


6518 6790 


7062 


7335 


3 




5 


7879 


8151 


8424 8696 


6968 


9240 


9512 


9785 


2030067 


2 




6 


2030601 


0873 


2031145 1417 


2031689 


1961 


2032233 


2505 


2777 


2 




7 


3321 


3593 


3865 4137 


4409 


4681 


4952 


5224 


5496 


2 




8 


6040 


6311 


6583 6855 


7126 


7398 


7670 


7941 


8213 


2 




9 


8756 


9028 


9299 9571 


9842 


,0114 

'2828 


2040385 


0657 


2040928 


2 




1600 


2041471 


1743 


2042014 2285 


2042557 ' 


3099 


3371 


3642 






1 


4185 


4456 


4727 4998 


5269 


5541 


5812 


6083 


6354 






2 


6896 


7167 


7438 7709 


7980 


8251 


8522 


8793 


9064 






3 


9606 


9877 


2050148 0419 


2050690 


0960 


2051231 


1502 


2051773 






4 


2052314 


2585 


2856 3127 


3397 


3663 


3939 


4209 


4480 






5 


5021 


5292 


5562 5833 


6103 


6374 


6644 


6915 


7185 






6 


7726 


7996 


8267 8537 


8807 


9078 


9348 


9618 


9889 







7 


2060429 


0699 


2060969 1240 


2061510 


1780 


2062050 


2320 


2062590 







8 


3131 


3401 


3671 3941 


4211 


4481 


4751 


5021 


5291 







9 


5830 


6100 


6370 6640 


6910 


7180 


7449 


7719 


7989 







1610 


8529 


8798 


9068 9338 


9607 


9877 


2070147 


0416 


2070686 







1 


2071225 


1495 


2071764 2034 


2072303 


2573 


2842 


3112 


3381 







2 


3920 


4189 


4459 4728 


4997 


5267 


5536 


5805 


6074 


269 




3 


6613 


6882 


7151 7421 


7690 


7959 


8228 


8497 


8766 


9 




4 


9304 


9573 


9842 .0111 


2080380 


0649 


2080918 


1187 


2081456 


9 




5 


2081994 


2263 


2082532/2801 


3070 


3338 


3607 


3876 


4145 


9 




6 


4682 


4951 


5220 5488 


5757 


6026 


6294 


6563 


6832 


9 




7 


7369 


7637 


7906 8174 


8443 


8711 


8980 


9248 


9517 


9 




8 


2090054 


0322 


2090590 0859 


2091127 


1395 


2091664 


1932 


2092200 


8 




9 


2737 


3005 


3273 3541 


3810 


4078 


4346 


4614 


4882 


8 






1 


2 


3 4 


5 


6 


7 


8 


9 



1 Table I.] 


LOGARITHMS 


OF NUMBERS 


FROM 1 TO 36,000. 


29 




Between 16200 = log." 


-1 4-2095150, and 10800 =. \ogr^ 4-2253093 






tem.i 1 


2 


3 


4 


5 


6 


T 


8 


9 


dif. 




1G20 


2095413 


5686 


2095954 


6222 


2096490 


6758 


2097026 


7294 


2097562 


268 




1 


8098 


8366 


8634 


8902 


9170 


9437 


9705 


9973 


2100241 


8 




2 


2100776 


1044 


2101312 


1579 


2101847 


2115 


2102382 


2650 


2918 


8 




3 


3453 


3720 


3988 


4255 


4523 


4790 


5058 


5325 


5593 


8 




4 


6128 


6395 


6662 


6930 


7197 


7464 


7732 


7999 


8266 


7 




5 


8301 


9068 


9335 


9603 


9870 


/0137 

^2808 


2110404 


0671 


2110938 


7 




6 


2111472 


1740 


2112007 


2274 


2112541 


3075 


3342 


3609 


7 




7 


4142 


4409 


4676 


4943 


5210 


5477 


5744 


6010 


6277 


7 




8 


6811 


7078 


7344 


7611 


7878 


8144 


8411 


8678 


8944 


7 




9 


9477 


9744 


2120011 


0277 


2120544 


0810 


2121077 


1343 


2121610 


7 




1630 


2122142 


2409 


2675 


2942 


3208 


3474 


3741 


4007 


4273 


6 




1 


4806 


5072 


5338 


5605 


5871 


6137 


6403 


6669 


6935 


6 




2 


7468 


7734 


8000 


8266 


8532 


8798 


9064 


9330 


9596 


6 




3 


2130128 


0394 


2130660 


0926 


2131191 


1457 


2131723 


1989 


2132255 


6 




4 


2786 


3052 


3318 


3584 


3849 


4115 


4381 


4646 


4912 


6 




5 


5443 


5709 


5974 


6240 


6505 


6771 


7037 


7302 


7568 


6 




6 


8098 


8364 


8629 


8895 


9160 


9425 


9691 


9956 


2140221 


5 




7 


2140752 


1017 


2141283 


1548 


2141813 


2078 


2142343 


2609 


2874 


5 




8 


3404 


3669 


3934 


4199 


4464 


4730 


4995 


5260 


5525 


5 




9 


€055 


6319 


6584 


6849 


7114 


7379 


7644 


7909 


8174 


5 




1640 


8703 


8668 


9233 


9498 


9762 


/0027 
'2673 


2150292 


0556 


2150821 


5 




1 


2151350 


1615 


2151880 


2144 


2152409 


2938 


3203 


3467 






2 


3996 


4260 


4525 


4789 


5054 


5318 


5583 


5847 


6111 






3 


6640 


6904 


7169 


7433 


7697 


7961 


8226 


8490 


8754 






4 


9282 


9546 


9811/0075 


2160339 


0603 


2160867 


1131 


2161395 






5 


2161923 


2187 


2162421' 


2715 


2979 


3243 


3507 


3771 


4034 






6 


4562 


4826 


5090 


5354 


5617 


5881 


6145 


6409 


6672 






7 


7200 


7463 


7727 


7991 


8254 


8518 


8781 


9045 


9309 






8 


9836/ 
2172470' 


0099 


2170363 


0626 


2170890 


1153 


2171416 


1680 


2171943 


3 




9 


2733 


2997 


3260 


3523 


3786 


4050 


4313 


4576 


3 




1650 


5103 


5366 


5629 


5892 


6155 


6418 


6682 


6945 


7208 


3 




1 


7734 


7997 


8260 


8523 


8786 


9049 


9312 


9575 


9838 


3 




2 


2180463 


0626 


2180889 


1152 


2181415 


1677 


2181940 


2203 


2182466 


3 




3 


2991 


3254 


3517 


3779 


4042 


4305 


4567 


4830 


5092 


3 




4 


5618 


5880 


6143 


6405 


6668 


6930 


7193 


7455 


7718 


2 




5 


8242 


8505 


8767 


9030 


9292 


9554 


9816 


/0079 
'2701 


2190341 


2 




6 


2190866 


1128 


2191390 


1652 


2191914 


2177 


2192439 


2963 


2 




7 


3487 


3749 


4011 


4273 


4535 


4797 


5059 


5351 


5583 


2 




8 


6107 


6369 


6631 


6893 


7155 


7417 


7678 


7940 


8202 


2 




9 


8726 


8987 


9249 


9511 


9773 


.0034 
^2650 


2200296 


0558 


2200819 


2 




1660 


2201342 


1604 


2201866 


2127 


2202389 ^ 


2912 


3173 


3435 


2 




1 


3958 


4219 


4481 


4742 


5003 


5265 


5526 


5788 


6049 






2 


6571 


6833 


7094 


7355 


7617 


7878 


8139 


8400 


8666 






3 


9184 


9445 


9706 


9967 


2210228 


0489 


2210750 


1011 


2211272 






4 


2211794 


2055 


2212316 


2577 


2838 


3099 


3360 


3621 


3882 






5 


4403 


4664 


4925 


5186 


5446 


5707 


5968 


6229 


6489 






6 


7011 


7271 


7532 


7793 


8053 


8314 


8574 


8835 


9095 






7 


9617 


9877 


2220138 


0398 


2220658 


0919 


2221179 


1440 


2221700 







8 


2222221 


2481 


2741 


3002 


3262 


3522 


3783 


4043 


4303 







9 


4824 


5084 


5344 


5604 


5864 


6124 


6384 


6645 


6905 







1670 


7425 


7685 


7945 


8205 


8465 


8725 


8985 


9245 


9505 







1 


2230024 


0284 


2230544 


0804 


2231064 


1324 


2231583 


1843 


2232103 







2 


2622 


2882 


3142 


3402 


3661 


3921 


4181 


4440 


4700 







3 


5219 


5479 


5738 


5998 


6257 


6517 


6776 


7036 


7295 







4 


7814 


8073 


8333 


8592 


8852 


9111 


9370 


9630 


9889 


259 




5 


2240407 


0667 


2240926 


1185 


2241444 


1704 


2241963 


2222 


2242481 


9 




6 


2999 


3258 


3517 


3777 


4036 


4295 


4554 


4813 


5072 


9 




7 


5590 


5849 


6107 


6366 


6625 


6884 


7143 


7402 


7661 


9 




8 


8178 


8437 


8696 


8955 


9213 


9472 


9731 


9990 


2250248 


9 




9 


2250766 


1024 


2251283 


1541 


2251800 


2059 


2252317 


2576 


2834 


9 






1 


2 


3 


^ 


5 


6 


T 


8 


9 


~ 








3^— 





















30 


LOGARITHMS OF NUMBERS 


FROM 1 TO 36,000. [Table i. i 


Between 16800 =:: log.-i 42253093, and 17400 = lo 


-.-1 4-2405492 




tens. 


f * 


2 


3^5 


6 


T 


8 


9 


dif. 


1680 


2253351 


3610 


2253868 4127 2254385 


4644 


2254902 


5160 


2255419 


258 


1 


5935 


6194 


6452 6710 6969 


7227 


7485 


7743 


8002 


8 


2 


8518 


8776 


9034 9293 9551 


9809 


2260067 


0325 


2260583 


8 


3 


12261099 


1357 


2261615 1873 2262131 


2389 


2647 


2905 


3163 


8 


4 


3679 


3937 


4194 4452 4710 


4968 


5226 


5484 


5741 


8 


5 


6257 


6515 


6772 7030 


7288 


7545 


7803 


8060 


8318 


8 


6 


8833 


9091 


9348 9606 


9863 


/0121 
'2695 


2270378 


0636 


2270893 


8 


7 


2271408 


1666 


2271923 2180 


2272438 


2953 


3210 


3467 




8 


3982 


4239 


4496 4753 


5011 


5268 


5525 


5782 


6039 




9 


6554 


6811 


7068 7325 


7582 


7839 


8096 


8353 


8610 




1690 


9124 


9381 


9638 9895 


2280152 


0409 


2280666 


0922 


2281179 




1 


2281693 


1950 


2282206 2463 


2720 


2977 


3233 


3490 


3747 




2 


4260 


4517 


4774 5030 


5287 


5543 


5800 


6057 


6313 




3 


6826 


7083 


.7339 7596 


7852 


8108 


8365 


8621 


8878 


6 


4 


9390 


9647 


9903 / 0159* 2290416 


0672^2290928 


1185 


2291441 


6 


5 


2291953 


2209 


2292466 ' 2722 


2978 


3234 


3490 


3746 


4002 


6 


6 


4515 


4771 


5027 5283 


5539 


5795 


6051 


6307 


6562 


6 


7 


7074 


7330 


7586 7842 


8098 


8354 


8609 


8865 


9121 


6 


8 


9633 


9888 


2300144 0400 


2300656 


0911 


2301167 


1423 


2301678 


6 


9 


2302189 


2445 


2701 2956 


3212 


3467 


3723 


3978 


4234 


6 


1700 


4745 


5000 


5256 5511 


5766 


6022 


6277 


6532 


6788 


5 


1 


7298 


7554 


7809 8064 


8320 


8575 


8830 


9085 


9340 


5 


2 


9851 


/0106 
'2656 


2310361 0616 


2310871 


1126 


2311331 


1636 


2311891 


5 


3 


2312401 


2911 3166 


3421 


3676 


3931 


4186 


4441 


5 


4 


4951 


5206 


5450 5715 5970 


6225 


6480 


6734 


6989 


5 


5 


7499 


7753 


8008 8263 


8517 


8772 


9026 


9281 


9536 


5 


6 


2320045 


0299 


2320554 0808 


2321063 


1317 


2321572 


1826 


2322081 


5 


7 


2590 


2844 


3098 3353 


3607 


3861 


4116 


4370 


4624 


4 


8 


5133 


5387 


5641 5896 


6150 


6404 


6658 


6912 


7166 


4 


9 


7675 


7929 


8183 8437 


6691 


8945 


9199 


9453 


9707 


4 


1710 


2330215 


0469 


2330723 0977 2331231 


1485 


2331739 


1992 


2332246 


4 


1 


2754 


3008 


3262 3515 


3769 


4023 


4277 


4530 


4784 


4 


2 


5291 


5545 


5799 6052 


6306 


6559 


6813 


7067 


7320 


4 


3 


7827 


8081 


8334 8588 


8841 


9095 


9343 


9601 


9855 


4 


4 2340362 


0615 


2340868 1122 2341375 


1628 


2341881 


2135 2342388 


3 


5 


2894 


3148 


3401 3654 


3907 


4160 


4414 


4667 


4920 


3 


6 


5426 


5679 


5932 6185 


6438 


6691 


6944 


7197 


7450 


3 


7 


7956 


8209 


8462 8715 


8967 


9220 


9473 


9726 


9979 


3 


8 


2350484 


0737 


2350990 1243 


2351495 


1748 


2352001 


2253 


2352506 


3 


9 


3011 


3264 


3517 3769 


4022 


4274 


4527 


4779 


5032 


3 


1720 


5537 


5789 


6042 6294 


6547 


6799 


7052 


7304 


7556 


2 


1 


8061 


8313 


8566 8818 


9070 


9323 


9575 


9827 


2360079 


2 


2 


2360584 


0836 


2361088 1340 


2361592 


1844 


2362097 


2349 


2601 


2 


3 


3105 


3357 


3609 3861 


4113 


4365 


4617 


4869 


5121 


2 


4 


5625 


5876 


6128 6380 


6632 


6834 7136 


7387 


7639 


2 


5 


8143 


8394 


8646 8898 


9150 


9401 


9653 


9905 


2370156 


2 


6 


2370660 


0911 


2371163 1414 


2371666 


1917 


2372169 


2420 


2672 


2 


7 


3175 


3426 


3678 3929 


4181 


4432 


4683 


4935 


5186 




8 


5689 


5940 


6191 6443 


6694 


6945 


7196 


7448 


7699 




9 


8201 


8452 


8703 8955 


9206 


9457 


9708 


9959 


2380210 




1730 


2380712 


0963 


2381214 1465 


2381716 


1967 


2382218 


2469 


2720 




1 


3222 


3472 


3723 3974 


4225 


4476 


4727 


4977 


5228 




2 


5730 


5980 


6231 6482 


6732 


6983 


7234 


7484 


7735 




3 


8236 


8487 


8737 8988 


9238 


9489 


9739 


9990 


2390240 





4 


2390741 


0992 


2391242 1493 


2391743 


1993 


2392244 


2494 


2744 





5 


3245 


34951 3746 3996 


4246 


4496 


4747 


4997 


5247 





6 


5747 


5998 


6248 6498 


6748 


6998 


7248 


7498 


7748 





7 


8248 


8498 


8748 8998 


9248 


9498 


9748 


9998 


2400248 





8 


2400748 


0997 


2401247 1497 


2401747 


1997 


2402247 


2496 


2746 





9 


3246 


3495 


3745 3995 


4244 


4494 


4744 


4993 


5243 








1 


3 


3 4 


5 


6 


T 


8 


9 






Table I.] 


LOGARITHMS 


OF NUMBERS 


FROM 1 TO 36,000. 


31 


Between 


17400 = log.- 


-1 4-2405492, and 18000 = log 


-1 4-2552725. 




tens. 


1 


2 


3 


4 


S 


6 


T 


8 9 

7489 2407738 


dif 


1740 


2405742 


5992 


2406241 


6491 


2406740 


6990 


2407239 


250 


1 


8237 


8487 


8736 


8985 


9235 


9484 


9734 


9983 2410232 


249 


2 


2410731 


0980 


2411229 


1479 


2411728 


1977 


2412226 


2476 2725 


9 


3 


3223 


3472 


3721 


3970 


4220 


4469 


4718 


4967 5216 


9 


4 


5714 


5963 


6212 


6461 


6710 


6959 


7208 


7457 7705 


9 


5 


8203 


8452 


8701 


8950 


9199 


9447 


9696 


9945 


2420194 


9 


6 


2420691 


0940 


2421189 


1437 


2421686 


1935 


2422183 


2432 


2680 


9 


J 7 


3178 


3426 


3675 


3923 


4172 


4420 


4669 


4917 


5166 


9 


8 


5663 


5911 


6160 


6408 


6656 


6905 


7153 


7401 


7650 


8 


9 


8146 


8395 


8643 


8891 


9139 


9388 


9636 


9884 


2430132 


8 


1750 


2430629 


0877 


2431125 


1373 


2431621 


1869 


2432117 


2365 


2613 


e 


1 


3109 


3357 


3605 


3853 


4101 


4349 


4597 


4845 


5093 


8 


2 


5589 


5837 


6085 


6332 


6580 


6828 


7076 


7324 


7571 


8 


3 


8067 


8315 


8562 


8810 


9058 


9305 


9553 


9801 


2440048 


8 


4 


2440543 


0791 


2441039 


1286 


2441534 


1781 


2442029 


2276 


2524 


8 


5 


3019 


32661 3514 
5740 5987 


3761 


4008 


4256 


4503 


4750 


4998 


7 


6 


5492 


6234 


6483 


6729 


6976 


7223 


7470 


7. 


7 


7965 


8212 


8459 


8706 


8953 


9200 


9448 


9695 


9942 


7 


8 


2450436 


0683 


2450930 


1177 


2451424 


1671 


2451918 


2165 


2452411 


7 


9 


2905 


3152 


3399 


3646 


3893 


4140 


4386 


4633 


4880 


7 


1760 


5373 


5620 


5867 


6114 


6360 


6607 


6854 


7100 


7347 


7 


1 


7840 


8087 


8333 


8580 


8826 


9073 


9320 


9566 


9813 


7 


2 


2460306 


0552 


2460798 


1045 


2461291 


1538 


2461 764 


2030 


2462277 


6 


3 


2769 


3016 


3262 


3508 


3755 


4001 


4247 


4493 


4740 


6 


4 


5232 


5478 


5724 


5970 


6217 


6463 


6709 


6955 


7201 


6 


5 


7693 


7939 


8185 


8431 


8677 


8923 


9169 


9415 


9661 


6 


6 


2470153 


0399 


2470645 


0891 


2471136 


1382 


2471628 


1874 


2472120 


6 


7 


2611 


2857 


3103 


3349 


3594 


3840 


4086 


4331 


4577 


6 


8 


5068 


5314 


5559 


5805 


6051 


6296 


6542 


6787 


7033 


6 


9 


7524 


7769 


8015 


8260 


8506 


8751 


8997 


9242 


9487 


5 


1770 


9978 /0223' 


2480469 


0714 


2480959 


1205 


2481450 


1695 


2481940 


5 

5 


1 


2482431 ' 


2676 


2921 


3166 


3412 


3657 


3902 


4147 


4392 


2 


4882 


5127 


5372 


5617 


5862 


6107 


6352 


6597 


6842 


5 


3 


7332 


7577 


7822 


8067 


8312 


8557 


8802 


9047 


9291 


5 


4 


9781 /0026' 


2490271 


0515 2490760 


1005 


2491249 


1494 


2491739 


5 
5 


5 


2492228^ 


2473 


2718 


2962 


3207 


3451 


3696 


3941 4185 


6 


4674 


4919 


5163 


5408 


5652 


5897 


6141 


6385 


6630 


4 


y 


7119 


7363 


7607 


7852 


8096 


8340 


8585 


8829 


9073 


4 


8 


9562 


9806 


2500050 


0294 


2500539 


0783 


2501027 


1271 


2501515 


4 


9 


2502004 


2248 


2492 


2736 


2980 


3224 


3468 


3712 


3956 


4 


1780 


4444 


4688 


4932 


5176 


5420 


5664 


5908 


6151 


6395 


4 


1 


6883 


7127 


7371 


7614 


7858 


8102 


8346 


8590 


8833 


4 


2 


9321 


9564 


9808/0052 


2510295 


0539 


2510783 


1026 


2511270 


4 


3 


2511757 


2001 


2512244 ' 


2488 


2713 


2975 


3218 


3462 


3705 


3 


4 


4192 


4435 


4679 


4922 


5166 


5409 


5652 


5896 


6139 


3 


5 


6625 


6869 


7112 


7355 


7599 


7842 


8085 


8328 


8571 


3 


6 


9058 


9301 


9544 


9787 


2520030 


0273 


2520516 


0759 


2521002 


3 


7 


2521489 


1732 


2521975 


2218 


2461 


2703 


2946 


3189 


3432 


3 


8 


3918 


4161 


4404 


4647 


4889 


5132 


5375 


5618 


5861 


3 


9 


6346 


6589 


6832 


7074 


7317 


7560 


7802 


8045 


8288 


3 


1790 


8773 


9016 


9258 


9501 


9743 


9986 


2530228 


0471 


2530713 


3 


1 


2531198 


1441 


2531683 


1926 


2532168 


2411 


2653 


2895 


3138 


2 


2 


3622 


3865 


4107 


4349 


4592 


4834 


5076 


5318 


5561 


2 


3 


6045 


6287 


6529 


6772 


7014 


7256 


7498 


7740 


7982 


2 


4 


8466 


8709 


8951 


9193 


9435 


9677 


9919 /016li2540403i 


2 


5 


2540886 


1128 


2541370 


1612 


2541854 


2096 2542338 ' 


2580 


2822 


2 


6 


3305 


3547 


3789 


4030 


4272 


4514 


4756 


4997 


52?!9 


2 


7 


5722 


5964 


6206 


6447 


6689 


6931 


7172 


7414 


7655 


2 


8 


8138 


8380 


8621 


8863 


9104 


9346 


9587 


9829 


2550070 


2 


9 


2550553 


0794 


2551036 


1277 


2551519 


1760 


2552001 


2242 


2484 


1 




1 


3 


3 


^ 


5 


6 


T 


8 


9 





32 


LOGARITHMS 


OF NUMBERS FROM 1 TO 36,000. [Table I. 


Between 18000 = log. ~ 


1 4'25527-25, and 18600 == log.-i 4-2695129. 


tens. 


1 


2 


3 


4 


r 5 6 


T 8 9 J dif.W 


1800 


2552966 


3208 


2553449 


3690 


2553931 4172 


2554414 4655 


2554896 


241 1 


1 


5378 


5619 


5860 


6102 


6343 6584 


6825 7066 


7307 


1 1 


2 


7789 


8030 


8271 


8512 


8753 8994 


9235 9475 


9716 


1 


3 


2560198 


0439 


2560680 


0921 


2561161 1402 


2561643 1884 


2562125 


1 


4 


2606 


2847 


3087 


3328 


3569 3810 


4050 4291 


4531 1 ir 


5 


5013 


5253 


5494 


5734 


5975 6215 


6456 6696 


6937 


1 


6 


7418 


7658 


7899 


8139 


8380 8620 


8860 9101 


9341 





7 


9822 


/0062 
2465 


2570302 


0543 


2570783 1023 


2571264 1504 


2571744 


' 


8 


2572224 


2705 


2945 


3185 3425 


3665 3905 


4146 





9 


4626 


4866 


5106 


5346 


5586 5826 


6066 6306 


6546 





1810 


7026 


7266 


7506 


7745 


7985 8225 


8465 8705 


8945 





1 


9424 


9664 


9904 fom 


2580383 0623 


2580863 1103 


2561342 





2 


2581822 


2061 


258230r 


2541 


2780 3020 


3259 3499 


3738 





3 


4218 


4457 


4697 


4936 


5176 5415 


5655 5894 


6133 





4 


6612 


6852 


7091 


7330 


7570 7809 


8048 82S8 


8527 


239 


5 


9006 


9245 


9484 


9723 


9963 /0202 
2592354 ' 2593 


2590441 0680 


2590919 


9 


6 


2591398 


1637 


2591876 


2115 


2832 3071 


3310 


9 


7 


3788 


4027 


4266 


4505 


4744 4983 


5222 5461 


5700 


9 


8 


6178 


6417 


6655 


6894 


7133 7372 


7611 7849 


BOSS 


9 


9 


8566 


8804 


9043 


9282 


9521 9759 


9998 /0237 
2602384 ' 2622 


2600475 


9 


1820 


2600952 


1191 


2601430 


1668 


2601907 2145 


2831 


9 


1 


3338 


3576 


3815 


4053 


4292 4530 


4769 5007 


5245 


8 


2 


5722 


5960 


6199 


6437 


6675 6914 


7152 7390 


7628 


8 


3 


8105 


8343 


8581 


8820 


9058 9296 


9534 9772 


2610010 


8 


4 


2610486 


0725 


2610963 


1201 


2611439 1677 


3611915 2153 


2391 


8 


5 


2867 


3105 


3343 


3580 


3818 4056 


4294 4532 


4770 


8 


6 


5246 


5483 


5721 


5959 


6197 6435 


6672 6910 


7148 


8 


7 


7623 


7861 


8099 


8336 


8574 8811 


9049 9287 


9524 


3 


8 


9999 


/0237 
'2612 


2620475 


0712 


2620950 1187 


2621425 1662 


2621900 


8 


9 


2622374 


2849 


3087 


3324 3562 


3799 4036 


4274 


7 


1830 


4748 


4986 


5223 


5460 


5697 5935 


6172 6409 


6646 




1 


7121 


4358 


7595 


7832 


8069 8306 


8543 8781 


9018 




2 


9492 


9729 


9966 


/0203 
2572 


2630440 0677 


2630914 1151 


2631388 




3 


2631862 


2098 


2632335 


2809 3046 


3283 3520 


3757 




4 


4230 


4467 


4704 


4940 


5177 5414 


5651 58S7 


6124 


7 


5 


6597 


6834 


7071 


7307 


7544 7780 


8017 8254 


8490 




6 


8963 


9200 


9436 


9673 


9909 /0146 
2642273 ^ 2510 


2640382 0619 


2640855 


6 


7 


2641328 


1564 


2641801 


2037 


2746 2982 


3219 


6 


8 


3691 


3928 


4164 


4400 


4636 4873 


5109 5345 


5581 


6 


9 


6053 


6290 


6526 


6762 


6998 7234 


7470 7706 


7944 


6 


1840 


8414 


8650 


8886 


9122 


9358 9594 


9830 /0066 
3652189 ' 2425 


2650302 


6 


1 


2650774 


1010 


2651246 


1481 


2651717 1953 


2660 


6 


2 


3132 


3368 


3604 


3839 


4075 4311 


4546 4782 


5018 


6 


3 


5489 


5725 


5960 


6196 


6431 6667 


6903 7138 


7374 


6 


4 


7845 


8080 


8316 


8551 


8787 9022 


9257 9493 


9728 


5 


5 


2660199 


0434 


2660670 


0905 


2661140 1376 


2661611 1846 


2662082 


5 


6 


2552 


2787 


3023 


3258 


3493 3728 


3963 4199 


4434 


5 


7 


4904 


5139 


5374 


5609 


5844 6080 


6315 6550 


6785 


5 


8 


7255 


7490 


7725 


7960 


8195 8429 


8664 8899 


9134 


5 


9 


9604 


9839 


2670074 


0309 


2670543 0778 


2671013 1248 


2671483 


5 


1850 


2671952 


2187 


2421 


2656 


2891 3126 


3360 3595 


3830 


5 


1 


4299 


4533 


4768 


5003 


5237 5472 


5706 5941 


6175 


5 


2 


6644 


6879 


7113 


7348 


7582 7817 


8051 8285 


8520 


4 


3 


8969 


9223 


9457 


9692 


9926 /0160 
2682268 ' 2503 


2680394 0629 


2680863 


4 


4 


2681332 


1566 


2681800 


2034 


2737 2971 


3205 


4 


5 


3673 


3907 


4141 


4376 


4610 4844 


5078 5312 


5546 


4 


6 


6014 


6248 


6482 


6716 


6950 7183 


7417 7651 


7885 


4 1 


7 


8353 


8587 


8821 


9054 


9288 9522 


9756 9990 


2690223 


4 


8 


2690691 


0925 


2691158 


1392 


2691626 1859 


2692093 2327 


2560 


4 


<' 


3028 


3261 


3495 


3728 


3962 4195 


4429 4662 


4896 


4 


1 


2 


3 


4 


5 6 


1 8 


9 


' 



Table i.] 


LOGARITHMS 


OF NUMBERS 


FROM 1 TO 36,000. 33 1 


Between 18600 = log." 


1 4-2695129, and 19200 = log.-i 4-2833012. 1 


tens. 


1 


2 


3 


4 


5 


6 


T 


8 


9 


dif. 


1860 


2695363 


5596 


2695830 


6063 


2696297 


6530 


2696764 


6997 


2697230 


233 


1 


7697 


7930 


8164 


8397 


8630 


8864 


9097 


9330 


9564 


3 


2 


2700030 


0263 


2700496 


0730 


2700963 


1196 


2701429 


1662 


2701895 


3 


3 


2362 


2595 


2828 


3061 


3294 


3527 


3760 


3993 


4226 


3 


4 


4692 


4925 


5158 


5391* 5624 


5857 


6090 


6323 


6555 


3 


5 


7021 


7254 


7487 


7720 


7953 


8185 


8418 


8651 


8884 


3 


6i 


9349 


9582 


9815, 
2712141/ 


0047 


2710280 


0513 


2710745 


0978 


2711211 


3 


7 


2711676 


1908 


2374 


2606 


2839 


3071 


3304 


3536 


3 


F 


4001 


4234 


4466 


4699 


4931 


5163 


5396 


5628 


5861 


2 


c 


6325 


6558 


6790 


7022 


7255 


7487 


7719 


7952 


8184 


2 


1870 


8648 


8881 


9113 


9345 


9577 


9809 


2720041 


0274 


2720506 


2 


1 


2720970 


1202 


2721434 


1666 


2721898 


2130 


2362 


2594 


2826 


2 


2 


3290 


3522 


3754 


3986 


4218 


4450 


4682 


4914 


5146 


2 


3 


5610 


5841 


6073 


6305 


6537 


6769 


7001 


7232 


7464 


2 


4 


7928 


8159 


8391 


8623 


8854 


9086 


9318 


9549 


9781 


2 


5 


2730244 


0476 


2730708 


0939 


2731171 


1402 


2731634 


1865 


2732097 


2 


6 


2560 


2791 


3023 


3254 


3486 


3717 


3949 


4180 


4411 




7 


4874 


5105 


5337 


5568 


5799 


6031 


6262 


6493 


6725 




8 


7187 


7416 


7650 


7881 


8112 


8343 


8574 


8806 


9037 




9 


9499 


9730 


9961 


.0192 
2502 


2740423 


0654 


2740885 


1116 


2741347 




1880 


2741809 


2040 


2742271 ^ 


2733 


2964 


3195 


3426 


3657 




1 


4119 


4350 


4581 


4811 


5042 


5273 


5504 


5735 


5965 




2 


6427 


6658 


6888 


7119 


7350 


7581 


7811 


8042 


8273 




3 


8734 


8964 


9195 


9426 


9656 


9887 


2750117 


0348 


2750578 




4 2751039 


1270 2751500 


1731 2751961 


2192 2422 


2653 


2883 





1 5 


3344 


3574 


3805 


4035 


4265 


4496 


4726 


4956 


5187 





6 


5647 


5877 


6108 


6338 


6568 


6798 


7028 


7259 


7489 





7 


7949 


8179 


8409 


8640 


8870 


9100 


9330 


9560 


9790 





8 


2760250 


0480 


2760710 


0940 


2761170 


1400 


2761630 


1860 


2762090 





9 


2549 


2779 


3009 


3239 


3469 


3699 


3929 


4158 


4388 





1890 


4848 


5078 


5307 


5537 


5767 


5997 


6226 


6456 


6686 





1 


7145 


7375 


7604 


7834 


8063 


8293 


8523 


8752 


8982 





2 


9441 


9670 


9900 


/0129 

2424 


2770359 


0588 


2770818 


1047 


2771277 





3 


2771736 


1965 


2772194 


2653 


2882 


3112 


3341 


3570 


229 


4 


4029 


4258 


4488 


4717 4946 


5175^ 5405 


5634 


5863^ 9 1 


5 


6321 


6550 


6780 


7009 


7238 


7467 


7696 


7925 


8154 


9 


6 


8612 


8841 


9070 


9299 


9528 


9757 


9986 


.0215 


2780444 


9 


7 


2760902 


1131 


2781360 


1589 


2781818 


2047 


2782276/2504 


2733 


9 


1 8 


3191 


3420 


3648 


3877 


4106 


4335 


4564 


4792 


5021 


9 


1 9 


5478 


5707 


5936 


6164 


6393 


6622 


6850 


7079 


7307 


9 


1900 


7765 


7993 


8222 


8450 


8679 


8907 


9136 


9364 


9593 


9 


1 


2790050 


0278 


2790506 


0735 


2790963 


1192 


2791420 


1648 


2791877 


8 


2 


2333 


2562 


2790 


3018 


3247 


3475 


3703 


3931 


4160 


8 


3 


4616 


4844 


5072 


5301 


5529 


5757 


5985 


6213 


6441 


8 


4 


6898 


7126 


7354 


7582 


7810 


8038 


8266 


8494 


8722 


8 


5 


9178 


9406 


9634 


9862 


2800090 


0317 


2800545 


0773 


28Q1001 


8 


6 


2801457 


1685 


2801912 


2140 


2368 


2596 


2824 


3051 


3279 


8 


7 


3735 


3962 


4190 


4418 


4645 


4873 


5101 


5328 


5556 


8 


8 


6011 


6239 


6467 


6694 


6922 


7149 


7377 


7604 


7832 


8 


9 


8287 


8514 


8742 


8969 


9197 


9424 


9651 


9879 


2810106 


7 


1910 


2810561 


0788 


2811016 


1243 


2811470 


1698 


2811925 


2152 


2380 


7 


1 


2834 


3061 


3289 


3516 


3743 


3970 


4197 


4425 


4652 


7 


2 


5106 


5333 


5560 


5787 


6014 


6242 


6469 


6696 


6923 


7 


3 


7377 


7604 


7831 


8058 


8285 


8512 


8739 


8966 


9192 


7 


4 


9646 


9873 


2820100 


0327 


2820554 


0781 


2821007 


1234 


2821461 


7 


5 


2821915 


2141 


2368 


2595 


2822 


3048 


3275 


3502 


3728 


7 


6 


4182 


4408 


4635 


4862 


5088 


5315 


5541 


5768 


5995 


7 


7 


6448 


6674 


6901 


7127 


7354 


7580 


7807 


8033 


6260 


7 


8 


8712 


8939 


9165 


9392 


9618 


9844 


2830071 


0297 


2830523 


6 


9 


2830976 


1202 


2831429 


1655 


2831881 


2107 


2334 


2560 


2786 


6 


* ' 


2 


3 


4 1 5 


6 


7 


8 


9 


.-._ 



34 LOGAKITHMS OF NUMBERS FROM 1 TO 36,000. [Table I. || 


Between 19200 = log.-' 4-2833012, and 19800 = log "i 4-296(5652. 1 


tens. 


1 2 


3 4 


5 6 


T 8 


9 


dif. 


1920 


2833238 3465 


2833691 3917 


2834143 4369 


2834595 4821 


2835048 


226 


1 


5500 5726 


5952 6178 


6404 6630 


6356 7082 


7308 


6 


2 


7760 7986 


8212 8438 


8663 8339 


9115 9341 


9567 


6 


3 


2840019 0245 


2840470 0696 


2840922 1148 


2341373 1599 


2841325 


6 


' 4 


2276 2502 


2728 2953 3179 3405' 


3630 3856 4082' 


6 


1 5 


4533 4759 


4984 5210 


5435 5661 


5886 6112 


6337 


6 


6 


6783 7014 


7239 7465 


7690 7916 


8141 8366 


8592 


6 


7 


9043 9268 


9493 9719 


9944 /0169 
2352196 ^ 2422 


2850394 0620 


2850845 


5 


8 


2851296 1521 


2851746 1971 


2647 2872 


3097 


5 


• 9 


3547 3773 


3993 4223 


4448 4673 


4893 5123 


5348 


5 


1930 


5798 6023 


6248 6473 


6698 6923 


7148 7373 


7598 


5 


1 


8048 8273 


8497 8722 


8947 9172 


9397 9622 


9846 


5 


2 


2860296 0521 


2860746 0970 


2361195 1420 


2861644 1869 


2862094 


5 


3 


2543 2763 


2993 3217 


3442 3666 


3391 4116 


4340 


5 


4 


4789 5014 


5238 5463 5687 5912' 


6136 6361 


6585 


5 


5 


7034 7259 


7433 7707 


7932 8156 


8381 8605 


8829 


4 


6 


9278 9502 


9726 9951 


2870175 0399 


2870624 0843 


2871072 


4 


7 


2871520 1745 


2871969 2193 


2417 2641 


2865 3090 


3314 


4 


8 


3762 3986 


4210 4434 


4658 4882 


5106 5330 


5554 


4 


9 


6002 6226 


6450 6674 


6398 7122 


7346 7570 


7793 


4 


1940 


8241 8465 


8689 8913 


9135 9360 


9584 9803 


2880032 


4 


1 1 


2880479 0703 


2330927 1150 


2881374 1593 


2881821 2045 


2269 


4 


2 


2716 2939 


3163 3387 


3610 3334 


4057 4281 


4504 


4 


3 


4952 5175 


5399 5622 


5845 6069 


6292 6516 


6739 


3 


4 


7186 7409 


7633 7856 3079 8303 


8526 8749 8973 


3 


i 5 


9419 9643 


9866 / 0089 
2892098 ' 2321 


2890312 0536 


2390759 0982 


2891205 


3 


6 


2891652 1875 


2544 2767 


2990 3213 


3436 


3 


7 


3883 4106 


4329 4552 


4775 4998 


5221 5444 


5667 


3 


6 


6112 6335 


6558 6781 


7004 7227 


7450 7673 


7896 


3 


9 


8341 8564 


8787 9010 


9232 9455 


9673 9901 


2900123 


3 


1950 


2900569 0792 


2901014 1237 


2901460 1682 


2901905 2127 


2350 


3 


1 


2795 3018 


3240 3463 


3686 3908 


4131 4353 


4576 


3 


2 


5021 5243 


5466 5638 


5910 6133 


6355 6578 


6800 


2 


3 


7245 7467 


7690 7912 


8134 8356 


8579 8301 


9023 


2 


4 


9468 9690 


9912/0135 2910357 0579 


2910801 1023 


2911245 


2 


5 


2911690 1912 


2912134 ' 2356 


2578 2800 


3022 3244 


3466 


2 


6 


3911 4133 


4355 4577 


4799 5020 


5242 5464 


5636 


2 


7 


6130 6352 


6574 6796 


7018 7240 


7461 7683 


7905 


2 


8 


8349 8570 


8792 9014 


9236 9453 


9679 9901 


2920123 


2 


9 


2920566 0788 


2921009 1231 


2921453 1674 


2921896 2118 


2339 


2 


1960 


2782 3004 


3225 3447 


3668 3390 


4111 4333 


4554 


2 


1 


4997 5219 


5440 5662 


5883 6105 


6326 6547 


6769 




2 


7211 7433 


7654 7875 


8097 8318 


8539 8760 


8982 




3 


9424 9645 


9867 /0083 
2932073 ' 2299 


2930309 0530 


2930751 0973 


2931194 




4 


2931636 1857 


2520 2741 


2962 3183 


3405 




5 


3847 4068 


4289 4510 


4730 4951 


5172 5393 


5614 




6 


6056 6277 


6498 6719 


6940 7160 


7381 7602 


7823 




7 


8264 8485 


8706 8927 


9147 9368 


9589 9810 


2940030 




8 


2940472 0692 


2940913 1134 


2941354 1575 


2941795 2016 


2237 




9 


2678 2898 


3119 3339 


3560 3730 


4001 4221 


4442 




1970 


4883 5103 


5324 5544 


5764 5985 


6205 6426 


6646 





1 


7087 7307 


7527 7748 


7968 8188 


8408 8629 


8849 





2 


9289 9510 


9730 9950 


2950170 0390 


2950610 0831 


2951051 





3 


2951491 1711 


2951931 2151 


2371 2591 


2811 3031 


3251 





41 3691 3911 


4131 4351 


4571 4791 


5011 5231 


5451 





c 


5891 6111 


6331 6550 


6770 6990 


7210 7430 


7650 





( 


8089 830C 


8529 8748 


8968 9188 


9408 9627 


9847 







2960286 0506 


2960726 0945 


2961165 1385 


2961604 1824 


2962043 


^ 


E 


2482 270: 


2922 3141 


3361 3580 


3800 4019 


4238 


219 


c 


4677 4897 


5116 5336 


5555 5774 


5994 6213 


6433 


^ 


^ 


1 2 


3 4 


5 6 I 7 8 


9 


_J 





Table I.] 


LOGARITHMS 


OF NUMBERS FROM 1 TO 36,000. 35 | 




Between 19800 = log." 


4-296G652, and 20400 = log 


.-> 43096302. j 




tens. 


1 


2 


3 


4 


5 G 


T - 


8 


9 


dif. 




1980 


2966871 


7091 


2967310 


7529 


2967748 7968 


2968187 


8406 


2968626 


219 




1 


9064 


9283 


9502 


9722 


9941 /0160 
2972132 '2351 


2970379 


0598 


2970817 


9 




2 


2971256 


1475 


2971694 


1913 


2570 


2789 


3008 


9 




3 


3446 


3665 


3884 


3103 


4322 4541 


4760 


4979 


5198 


9 




4 


3636 


5854 


6073 


6292 6511 6730* 


6949 


7168 


7386 9 il 




5 


7824 


8043 


8261 


8480 


8699 8918 


9136 


9355 


9574 


9 




6 


29S0011 


0230 


2980448 


0667 


2980886 1104 


2981323 


1542 


2981760 


9 




7 


2197 


2416 


2634 


2853 


3071 3290 


3508 


3727 


3945 


8 




8 


4382 


4601 


4819 


5038 


5256 5474 


5693 


5911 


6129 


8 




9 


6566 


0785 


7003 


7221 


7439 7658 


7876 


8094 


8313 


8 




1990 


8749 


8967 


9185 


9404 


9622 9840 


2990058 


0276 


2990494 


8 




1 


2990931 


1149 


2991367 


1585 


2991803 2021 


2239 


2457 


2675 


8 




2 


3111 


3329 


3547 


3765 


3983 4201 


4419 


4637 


4855 


8 




3 


5291 


5509 


5727 


5945 


6162 6380 


6598 


6816 


7034 


8 




4 


7469 


7687 7905 


8123 


8340 8558 


8776 


8994 


9211 


8 




5 


9647 


9864 


3000082 


0300 


3000517 0735 


3000953 


1170 


3001388 


8 




6 


3001823 


2041 


2258 


2476 


2693 2911 


3128 


3346 


3563 


8 




7 


3998 


4216 


4433 


4650 


4868 5085 


5303 


5520 


5737 


7 




8 


6172 


6390 


6607 


6824 


7042 7259 


7476 


7693 


7911 


7 




9 


8345 


8562 


8780 


8997 


9214 9431 


9648 


9866 


3010083 


7 




2000 


3010517 


0734 


3010951 


1168 


3011386 1603 


3011820 


2037 


2254 


7 




1 


2688 


2905 


3122 


3339 


3556 3773 


3990 


4207 


4424 


7 




2 


4S58 


5075 


5291 


5508 


5725 5942 


6159 


6376 


6593 


7 




3 


7026 


7243 


7460 


7677 


7893 81 IT) 


8327 


8544 


8760 


7 




4 


9194 


9411 


9627 


9844 


3020061 0277 


3020494 


0711 '3020927 


7 




5 


3021360 


1577 


3021794 


2010 


2227 2443 


2660 


2876 


3093 


7 




6 


3526 


3742 


3959 


4175 


4392 4608 


4825 


5014 


5257 


6 




7 


5690 


5906 


6123 


6339 


6556 6772 


6988 


7204 


7421 


6 




8 


7853 


8070 


8286 


8502 


8718 8935 


9151 


9367 


9583 


6 




9 


3030016 


0232 


3030448 


0664 


3030880 1096 


3031312 


1528 


3031745 


6 




2010 


2177 


2393 


2609 


2825 


3041 3257 


3473 


3689 


3905 


6 




1 


4337 


4553 


4769 


4984 


5200 5416 


5632 


5848 


6064 


6 




2 


6496 


6711 


6927 


7143 


7359 7575 


7790 


8006 


8222 


6 




3 


8653 


8869 


9085 


9301 


9516 9732 


9948 


,0163 


3040379 


6 




4 


3040810 


1026 


3041242 


1457 


3041673 1888 


3042104 


/2319 


2535 


6 




5 


2966 


3182 


3397 


3613 


3828 4043 


4259 


4474 


4690 


5 




6 


5121 


5336 


5552 


5767 


5982 6198 


6413 


6628 


6844 


5 




7 


7274 


7490 


7705 


7920 


8135 8351 


8566 


8781 


8996 


5 




8 


9427 


9642 


9857 


/0072 
'2224 


3050288 0503 


3050718 


0933 


3051148 


5 




9 


3051578 


1793 


3052008 


2439 2654 


2869 


3084 


3299 


5 




2020 


3729 


3944 


4159 


4374 


4589 4803 


5018 


5233 


5448 


5 




1 


5878 


6093 


6308 


6523 


6737 6952 


7167 


7382 


7597 


5 




2 


8026 


8241 


8456 


8671 


8885 9100 


9315 


9529 


9744 


5 




3 


3060174 


0388 


3060603 


0817 


3061032 1247 


3061461 


1676 


3061891 


5 




4 


2320 


2534 


2749 


2963 


3178 3392 


3607 


3821 


4036 


5 




5 


4465 


4679 


4894 


5108 


5322 5537 


5751 


5966 


6180 


4 




6 


6609 


6823 


7037 


7252 


7466 7680 


7895 


8109 


8323 


4 




7 


8752 


8966 


9180 


9394 


9609 9823 


3070037 


0251 


3070465 


4 




8 


3070894 


1108 


3071322 


1536 


3071750 1964 


2178 


2392 


2606 


4 




9 


3035 


3249 


3463 


3677 


3891 4105 


4319 


4532 


4746 


4 




2030 


5174 


5388 


5602 


5816 


6030 6244 


6458 


6672 


6885 


4 




1 


7313 


7527 


7741 


7954 


8168 8382 


8596 


8810 


9023 


4 




2 


9451 


9664 


9878 


.0092 


3080306 0519 


3080733 


0947 


3081160 


4 




3 


3081587 


1801 


3082015/ 2228 


2442 2655 


2869 


3082 


3296 


4 




4 


3723 


3936 


4150 


4363 


4577 4790. 


5004 


5217 


5431 


4 




5 


5858 


6071 


6284 


6498 


6711 6924 


7138 


7351 


7564 


3 




6 


7991 


8204 


8418 


8631 


8844 9057 


9271 


9484 


9697 


3 




7 


3090123 


0337 


3090550 


0763 


3090976 1189 


3091402 


1616 


3091829 


3 




8 


2255 


2468 


2681 


2894 


3107 3320 


3533 


3746 


3959 


3 




9 


4385 


4598 


4811 


5024 


5237 5450 


5663 


5876 


6089 


3 






1 


2 


3 


4 


.5 6 


t 


8 


9 





36 


LOGARITHMS 


OF NUMBERS FROM 1 TO 36,000. [Table 1. II 


Between 30400 ^ log." 


-1 4-3096302, and 21000 = log 


-1 4-3222193. 1 


tCTlS. 


1 


2 


3 


4 


o 


t> t 


8 


9 


dif. 


2040 


3096515 


6727 


3096940 


7153 


3097366 


7579 3097792 


8004 


3098217 


213 


i 


8643 


8856 


9008 


9231 


9494 


9707 9919 


0132 


3100345 


3 


: 2 


3100770 


0933 


3101195 


1408 


3101621 


1833 


3102046 / 


2258 


2471 


3 


'■ 3 


2896 


3109 


3321 


3534 


3746 


3959 


4171 


4334 


4596 


3 


1 4 


5021 


5234 


5446 


5659 5371 


6084 


6296 


6503 


6721 


3 


1 51 7145 


7358 


7570 


7783 


7995 


8207 


8419 


8632 


8844 


2 


1 6 


9269 


9481 


9693 


9905 


3110117 


0330 


3110542 


0754 


3110966 


2 


1 7 


3111391 


1603 


3111815 


2027 


2239 


2451 


2663 


2875 


3037 


2 


1 8 


3512 


3724 


3936 


4148 


4360 


4572 


4784 


4996 


5208 


2 


9 


5632 


5843 


6055 


6267 


6479 


6691 


6903 


7115 


7327 


2 


2050 


7750 


7962 


8174 


8336 


8593 


8310 


9021 


9233 


9445 


2 


1 


9868 / 0080 


3120292 


0504 


3120715 


0927 


3121139 


1350 


3121562 


2 


2 


31219S5' 


2197 


2403 


2620 


2832 


3043 


3255 


3466 


3678 


2 


3 


4101 


4313 


4524 


4736 


4947 


5159 


5370 


5581 


5793 


2 


4 


6216 


6427 


6639 


6850 7061 


7273 


7434 


7696 


7907 




5 


8330 


8541 


8752 


8964 


9175 


9336 


9597 


9809 


3130020 




6 


3130442 


0654 


3130365 


1076 


3131237 


1493 


3131709 


1921 


2132 




7 


2554 


2765 


2976 


3187 


3398 


3610 


3321 


4032 


4243 




8 


4665 


4876 


5087 


5298 


5509 


5720 


5931 


6142 


6353 




9 


6774 


6935 


7196 


7407 


7613 


7829 


8040 


8251 


8461 




2060 


8383 


9094 


9305 


9515 


9726 


9937 


3140148 


0353 


3140569 




1 


3140991 


1201 


3141412 


1623 


3141833 


2044 


2255 


2465 


2676 




2 


3097 


3308 


3513 


3729 


3940 


4150 


4361 


4571 


4732 




3 


5203 


5413 


5624 


5834 


6045 


6255 


6466 


6676 


6837 




4 


7307 


7518 


7723 


7939 8149 


8359 8570 


8780 


8990 





5 


9411 


9621 


9831 / 00421 


3150252 


0462 


3150672 


0883 


3151093 





6 


3151513 


1724 


3151934 ^ 


2144 


2354 


2564 


2774 


2985 


3195 





7 


3615 


3825 


4035 


4245 


4455 


4665 


4875 


5085 


5295 





8 


5715 


5925 


6135 


6345 


6555 


6765 


6975 


7185 


7395 





9 


7815 


8025 


8235 


8444 


8654 


8864 


9074 


9284 


9494 





2070 


9913/0123 


3160333 


0543 


3160752 


0962 


3161172 


1382 


3161591 





1 


3162011 ' 


2220 


2430 


2640 


2849 


3059 


3269 


3478 


3688 





2 


4107 


4317 


4526 


4736 


4945 


5155 


5364 


5574 


5784 





3 


6203 


6412 


6621 


6831 


7040 


7250 


7459 


7669 


7878 


209 


4 


8297 


8506 8716 


8925 


9134 


9344 9553 


9762 9972 


9 


5 


3170390 


0600 


3170809 


1018 


3171227 


1437 


3171646 


1855 


3172064 


9 


6 


2483 


2692 


2901 


3110 


3319 


3528 


3738 


3947 


4156 


9 


7 


4574 


4783 


4992 


5201 


5410 


5619 


5828 


6037 


6-246 


9 


8 


6664 


6873 


7082 


7291 


7500 


7709 


7918 


81-27 


8336 


9 


9 


8754 


8963 


9172 


9380 


9589 


9798 


3180007 


0216 


31S0425 


9 


2080 


3180842 


1051 


3181260 


1468 


3181677 


1836 


2095 


2303 


2512 


9 


1 


2929 


3138 


3347 


3556 


3764 


3973 


4181 


4390 


4599 


9 


2 


5016 


5224 


5433 


5642 


5850 


6059 


6267 


6476 


6684 


9 


3 


7101 


7310 


7518 


7727 


7935 


8143 


8352 


8560 


8769 


8 


4 


9136 


9394 


9602 


9811 


3190019 


0227 


3190436 


0644 


3190852 


8 


5 


3191269 


1477 


3191685 


1894 


2102 


2310 


2518 


2727 


2935 


8 


6 


3351 


3559 


3768 


3976 


4184 


4392 


4600 


4808 


5016 


8 


7 


5433 


5641 


5849 


6057 


6265 


6473 


6681 


6889 


7097 


8 


8 


7513 


7721 


7929 


8137 


8345 


8553 


8761 


8969 


9176 


8 


9 


9592 


9800 


3200008 


0216 


3200424 


0632 


3200839 


1047 


3201255 


8 


2090 


3201671 


1878 


208G 


2294 


2502 


2709 


2917 


3125 


3333 


8 


1 


3748 


3956 


4163 


4371 


4579 


4786 


4994 


5202 


5409 


8 


2 


5624 


6032 


6240 


6447 


6655 


6862 


7070 


7277 


7485 


8 


3 


7900 


8107 


8315 


8522 


8730 


8937 


9145 


9352 


9559 




4 


9974 


,0182 
/2255 


3210389 


0596 


3210804 


1011 


3211218 


1426 


3211633 




5 


3212048 


2462 


2669 


2877 


3084 


3291 


3498 


3706 




6 


4120 


4327 


4534 


4742 


4949 


5156 


5363 


5570 


5777 




1 I 


6191 


6398 


6606 


6813 


7020 


7227 


7434 


7641 


7848 




1 S 


8262 


8469 


8676 


8883 


9090 


9297 


9504 


9711 


9917 




1 ^ 


3220331 


0538 


3220745 


0952 


3221159 


1366 


3221572 


1779 


3221986 




lU 


1 


2 


3 


4 1 5 


6 


7 


8 


9 






Table I.] 


LOGARITHMS OF NUMBERS FROM 1 TO 36,000. 37 ll 


Between 21000 = log.^i 4-3222193, and 21600 = log.-i 4-3344538. || 


tens. 


1 


z 

2607 


3 4r 


5 


6 


T 8 


9 


dif 


2100 


3222400 


3222813 3020 


3223227 


3434 


3223640 3847 


3224054 


207 


1 


4467 


4674 


4881 5087 


5294 


5501 


5707 5914 


6121 


7 


2 


6534 


6740 


6947 7153 


7360 


7567 


7773 7980 


8186 


7 


3 


8599 


8806 


9012 9219 


9425 


9632 


9838 /0045 
3231902 / 2108 


3230251 


6 


4 


3230664 


0870 


3231077 1283 


3231489 


1696 


2315 


6 


5 


2727 


2934 


3140 3346 


3552 


3759 


3965 4171 


4377 


6 


6 


4790 


4996 


5202 5408 


5615 


5821 


6027 6233 


6439 


6 


7 


6851 


7058 


7264 7470 


7676 


7882 


8088 8294 


8500 


6 


8 


8912 


9118 


9324 9530 


9736 


9942 


3240148 0354 


3240560 


6 


9 


3240972 


1178 


3241384 1589 


3241795 


2001 


2207 2413 


2619 


6 


2110 


3030 


3236 


3442 3648 


3854 


4059 


4265 4471 


4677 


6 


1 


5088 


5294 


5499 5705 


5911 


6117 


6322 6528 


6734 


6 


2 


7145 


7350 


7556 7762 


7967 


8173 


8378 8584 


8789 


6 


3 


9201 


9406 


9612 9817 


3250023 


0228 


3250433 0639 


3250844 


5 


4 


3251255 


1461 


3251666 1872 


2077 


2282 


2488 2693 


2898 


5 


5 


3309 


3514 


3720 3925 


4130 


4336 


4541 4746 


4951 


5 


6 


5362 


5567 


5772 5978 


6183 


6388 


6593 6798 


7003 


5 


7 


7414 


7619 


7824 8029 


8234 


8439 


8644 8849 


9055 


5 


8 


9465 


9670 


9875 /0080 
3261924 ' 2129 


3260285 


0490 


3260695 0900 


3261105 


5 


9 


3261515 


1719 


2334 


2539 


2744 2949 


3154 


5 


2120 


3563 


3768 


3973 4178 


4383 


4588 


4792 4997 


5202 


5 


1 


5611 


5816 


6021 6226 


6430 


6635 


6840 7044 


7249 


5 


2 


7658 


7863 


8068 8272 


8477 


8682 


8886 9091 


9295 


5 


3 


9705 


9909 


3270114 0318 


3270523 


0727 


3270932 1136 


3271341 


5 


4 


3271750 


1954 


2158 2363 2567 


2772 


2976 3181 


3385 




5 


3794 


3998 


4202 4407 


4611 


4815 


5020 5224 


5428 




6 


5837 


6041 


6245 6450 


6654 


6858 


7062 7267 


7471 




7 


7879 


8083 


8287 8492 


8696 


8900 


9104 9308 


9512 




8 


9920 


/0124 


3280328 0533 


3280737 


0941 


3281145 1349 


3281553 




9 


3231961 / 2165 


2369 2572 


2776 


2980 


3184 3388 


3592 




2130 


4000 


4204 


4408 4612 


4815 


5019 


5223 5427 


5631 




1 


6038 


6242 


6446 8650 


6853 


7057 


7261 7465 


7668 




2 


8076 


8279 


8483 8687 


8890 


9094 


9298 9501 


9705 




3 


3290112 


0316 


3290519 0723 


3290926 


1130 


3291334 1537 


3291741 




4 


2148 


2351 


2555 2758 


2962 


3165 


3369 3572 


3775' 3 II 


5 


4182 


4386 


4589 4792 


4996 


5199 


5402 5606 


5809 


3 


6 


6216 


6419 


6622 6826 


7029 


7232 


7436 7639 


7842 


3 


7 


8248 


8452 


8655 8858 


9061 


9264 


9468 9671 


9874 


3 


8 


3300280 


0483 


3300686 0889 


3301093 


1296 


3301499 1702 


3301905 


3 


9 


2311 


2514 


2717 2920 


3123 


3326 


3529 3732 


3935 


3 


2140 


4341 


4544 


4747 4949 


5152 


5355 


5558 5761 


5964 


3 


1 


6370 


6572 


6775 6978 


7181 


7384 


7586 7789 


7992 


3 


2 


8397 


8600 


8803 9006 


9208 


9411 


9614 9816 


3310019 


3 


3 


3310424 


0627 


3310830 1032 


3311235 


1437 


3311640 1843 


2045 


3 


4 


2450 


2653 


2855 3058 


3261 


3463 


3666 3868 


4070 


2 


5 


4475 


4678 


4880 5083 


5285 


5488 


5690 5892 


6095 


2 


6 


6500 


6702 


6904 7107 


7309 


7511 


7714 7916 


8118 


2 


7 


8523 


8725 


8927 9129 


9332 


9534 


9736 9938 


3320141 


2 


8 


3320545 


0747 


3320949 1151 


3321354 


1556 


3321758 1960 


2162 


2 


9 


2566 


2768 


2970 3172 


3374 


3577 


3779 3981 


4183 


2 


2150 


45B7 


4789 


4991 5193 


5394 


5596 


5798 6000 


6202 


3 


1 


6606 


6808 


7010 7212 


7414 


7615 


7817 8019 


8221 


3 


2 


8624 


8826 


9028 9230 


9432 


9633 


9835 /0037 
3331852 1 2054 


3330239 


3 


3 


3330642 


0844 


3331045 1247 


3331449 


1650 


2255 


2 


4 


2659 


2860 


3062 3263 


3465 


3667 


3868 4070 


4271 




5 


4674 


4876 


5077 5279 


5480 


5682 


5883 6085 


6286 




6 


6689 


6890 


7092 7293 


7495 


7696 


7897 8099 


8300 




7 


8703 


8904 


9105 9307 


9508 


9709 


9911/0112 
3341923' 2124 


3340313 




8 


3340716 


0917 


3341118 1319 


3341521 


1722 


2325 




9 


2728 


2929 


3130 3331 


3532 


3733 


3934 4135 


4336 






1 


2 


3 4 


5 


6 


T 8 


-^-Ml 



38 LOGARITHMS 


OF NUMBERS 


FROM 1 TO 36,000. [Table i. 


Between 21600 = log. ~ 


1 4-3344538, and 22200 = log. -i 4-3463530. [ 


tens. 


? 1 a 


3 


^1 


5 


6 


T 


8 


9 


dif. 


2160 


3344739 4940 


3345141 


5342 


3345543 


5744 


3345945 


6146 


3346347 


201 


1 


6749 6950 


7151 


7351 


7552 


7753 


7954 


8155 


8356 


1 


2 


8758 8959 


9159 


9360 


9561 


9762 


9963 


/0164 
^2171 


3350364 


1 


3 


3350766 0967 


3351168 


1368 


3351569 


1770 


3351970 


2372 


1 


4 


2773 2974 


3175 


3375 


3576 


3777 


3977 


4178 


4378 


1 


1 5 


4780 4980 


5181 


5381 


5582 


5782 


5983 


6183 


6384 


1 


1 6 


6785 6986 


7186 


7386 


7587 


7787 


7988 


8188 


8389 


200 


7 


8790 8990 
3360793 0993 


9190 


9391 


9591 


9791 


9992 


/0192 
^2195 


3360392 





8 


3361194 


1394 


3361594 


1795 


3361995 


2395 





9 


2796 2996 


3196 


3396 


3597 


3797 


3997 


4197 


4397 





^2170 


4797 4998 


5198 


5398 


5598 


5798 


5998 


6198 


6398 





1 1 


6798 6998 


7198 


7398 


7598 


7798 


7998 


8198 


8398 


Oi 


2 


8798 8998 


9198 


939S 


9598 


9798 


9998 


/0198 
^2196 


3370397 


! 


3 


13370797 0997 


3371197 


1397 


3371596 


1796 


3371996 


2396 


' 


4! 2795 2995 


3195 


3394 


3594 


3794 


3994 


4193 


4393 





5 


4792 4992 


5192 


5391 


5591 


5791 


5990 


6190 


6389 





1 6 


6788 6988 


7188 


7387 


7587 


7786 


7986 


8185 


8385 


' 


; 7 


8784 8983 


9183 


9382 


9582 


9781 


9981 


/0180 
' 2174 


3380379 


199; 


8 


3380778 0978 


3381177 


1376 


3381576 


1775 


3381974 


2373 


9 1 


9 


2772 2971 


3170 


3369 


3569 


3768 


3967 


4166 


4366 


9 j 


|2180 


4764 4963 


5163 


5362 


5561 


5760 


5959 


6158 


6358 


9 


1 


6756 6955 


7154 


7353 


7552 


7751 


7950 


8149 


834R 


9 


2 


8746 8946 


9145 


9344 


9543 


9742 


9940 


/0139 
'2129 


3390338 


9 1 


3 


3390736 0935 


3391134 


1333 


3391532 


1731 


3391930 


2327 


9 


4 


2725 2924^ 3123 


3322 


3520 


3719 


3918 


4117 


4316 


9 


5 


4713 4912 


5111 


5309 


5508 


5707 


5906 


6104 


6303 


9 


6 


6700 6899 


7098 


7296 


7495 


7693 


7892 


8091 


82S9 


9 


7 


8686 8885 


9084 


9282 


9481 


9679 


9878 


/0076 
'2061 


3400275 


9 


8 


3400672 0870 


3401069 


1267 


3401466 


1664 


3401862 


2259 


198 


1 9 


2656 2854 


3053 


3251 


3449 


3648 


3846 


4045 


4243 


8 


, 2190 


4639 4838 


5036 


5234 


5433 


5631 


5829 


6027 


6226 


8 


1 


6622 6820 


7018 


7217 


7415 


7613 


7811 


8009 


8207 


8 


2 


8604 8802 


9000 


9198 


9396 


9594 


9792 


9990 


3410188 


8 


3 


3410584 0782 


3410930 


1178 


3411376 


1574 


3411772 


1970 


2168 


8 


4 


2564 2762 2960 


3158 


3356 


3554 


3752 


3950 


4147 


8 


5 


4543 4741 


4939 


5137 


5334 


5532 


5730 


5928 


6126 


8 


6 


6521 6719 


6917 


7114 


7312 


7510 


7708 


7905 


8103 


8 


7 


8498 8696 


8894 


9091 


9289 


9486 


9684 


9882 


3420079 


8 


8 


3420474 0672 


3420870 


1067 


3421265 


1462 


3421660 


1857 


2055 


8 


9 


2450 2647 


2845 


3042 


3240 


3437 


3635 


3832 


4029 


197 


2200 


4424 4622 


4819 


5016 


, 5214 


5411 


5608 


5806 


6003 


7 


1 


6398 6595 


6792 


6990 


7187 


7384 


7581 


7779 


7976 


7 


2 


8370 8568 


8765 


8962 


9159 


9356 


9554 


9751 


9948 


7 


3 


3430342 0539 


3430736 


0933 


3431131 


1328 


3431525 


1722 


3431919 


7 


4 


2313 2510 


2707 


2904 


3101 


3-298 


3495 


3692 


3889 


7 


5 


4283 4480 


4677 


4874 


5071 


5268 


5464 


5661 


5858 


7 


6 


6252 6449 


6646 


6842 


7039 


7236 


7433 


7630 


7827 


7 


7 


8220 8417 


8614 


8810 


9007 


9204 


9401 


9597 


9794 


7 


8 


3440187 0384 


3440581 


0777 


3440974 


1171 


3441367 


1564 


3441761 


7 


9 


2154 2350 


2547 


2743 


2940 


3137 


3333 


3530 


37-26 


7 


2210 


4119 4316 


4512 


4709 


4905 


5102 


5298 


5495 


5691 


196 


1 


6084 6280 


6477 


6673 


6869 


7066 


7262 


7459 


7655 


6 


2 


8084 8244 


8440 


8636 


8833 


9029 


9225 


9422 


9618 


6 


3 


3450010 0207 


3450403 


0599 


3450795 


0991 


3451188 


1384 


3451580 


6 


4 


1972 2168 


2365 


2561 


2757 


2953 


3149 


3345 


3541 


6 


5 


3933 4129 


4325 


4522 


4718 


4914 


5110 


5306 


5502 


6 


6 


5894 6090 


6285 


6481 


6677 


6873 


7069 


7265 


7461 


6 


7 


7853 8049 


8245 


8440 


8636 


8832 


9028 


9224 


9420 


6 


8 


9811 /0007 
3461769 ' 1964 


3460203 


0399 


3460594 


0790 


3460986 


1182 


3461377 


6 


9 


2160 


2356 


2551 


2747 


2943 


3138 


3334 


6 




1 2 


3 


4- 


5 


6 


T 


8 


« ^ 



Table I.] 


LOGARITHMS OF NUMBERS FROM 1 TO 36,000. 39 11 


Between 22200 = log.-i 4-3463530, and 22800 = log 


-1 4-3579348. 1 


tens. 1 


2 


3 4 


5 


6 


T 


8 


9 


dif. 


2220 


3463725 


3921 


3464117 4312 


3464508 


4703 


3464899 


5094 


3465290 


196 


1 


5681 


5877 


6072 6268 


6463 


6659 


6854 


7050 


7245 


195 


2 


7636 


7831 


8027 8222 


8418 


8613 


8808 


9004 


9199 


5 


3 


9590 


9785 


9981 /0176 
3471934 / 2129 


3470371 


0567 


3470762 


0957 


3471153 


5 


4 


3471543 


1738 


2324 


2519 


2715 


2910 


3105 5 II 


5 


3495 


3691 


3886 4081 


4276 


4471 


4666 


4861 


5056 


5 


6 


5447 


5642 


5837 6032 


6227 


6422 


6617 


6812 


7007 


5 


7 


7397 


7592 


7787 7982 


8177 


8372 


8567 


8762 


8957 


5 


8 


9347 


9542 


9737 9931 


3480126 


0321 


3480516 


0711 


3480906 


5 


9 


3481296 


1490 


3481685 1880 


2075 


2270 


2464 


2659 


2854 


5 


2230 


3243 


3438 


3633 3828 


4022 


4217 


4412 


4606 


4801 


5 


1 


5190 


5385 


5580 5774 


5969 


6164 


6358 


6553 


6747 


5 


2 


7136 


7331 


7526 7720 


7915 


8109 


8304 


8498 


8693 


5 


3 


9082 


9276 


9471 9665 


9860 


/0054 


3490248 


0443 


3490637 


194 


4 


3491026 


1220 


3491415 1609 


3491804 / 1998' 


2192 


2387 


2581 4 11 


5 


2970 


3164 


3358 3552 


3747 


3941 


4135 


4330 


4524 


4 


6 


4912 


5106 


5301 5495 


5689 


5883 


6077 


6272 


6466 


4 


7 


6854 


7048 


7242 7436 


7630 


7825 


8019 


8213 


8407 


4 


8 


8795 


8989 


9183 9377 


9571 


9765 


9959 


/0153 


3500347 


4 


9 


3500735 


0929 


3501123 1317 


3501511 


1705 


3501898 1 2092 


2286 


4 


2240 


2674 


2868 


3062 3256 


3449 


3643 


3837 


4031 


4225 


4 


1 


4612 


4806 


5000 5194 


5387 


5581 


5775 


5969 


6162 


4 


2 


6550 


6743 


6937 7131 


7325 


7518 


7712 


7905 


8099 


4 


3 


8486 


8680 


8874 9067 


9261 


9454 


9648 


9841 


3510035 


4 


4 3510422 


0616 


3510809 1003 3511196 


1390 


3511583 


1777 1970*193 || 


5 


2357 


2550 


2744 2937 


3131 


3324 


3517 


3711 


3905 


3 


6 


4291 


4484 


4678 4871 


5064 


5258 


5451 


5644 


5837 


3 


7 


6224 


6417 


6611 6804 


6997 


7190 


7383 


7577 


7770 


3 


8 


8156 


8349 


8543 8736 


8929 


9122 


9315 


9508 


9701 


3 


9 


3520088 


0281 


3520474 0667 


3520860 


1053 


3521246 


1439 


3521632 


3 


2250 


2018 


2211 


2404 2597 


2790 


2983 


3176 


3369 


3562 


3 


1 


3948 


4141 


4334 4527 


4720 


4912 


5105 


5298 


5491 


3 


2 


5877 


6070 


6262 6455 


6648 


6841 


7034 


7226 


7419 


3 


3 


7805 


7997 


8190 8383 


8576 


8768 


8961 


9154 


9346 


3 


4 


«732 


9924 3530117 0310 '3530502 


0695 3530888 


1080 3531273 


3 


5 


3531658 


1851 


2043 2236 


2428 


2621 


2813 


3006 


3198 


3 


6 


3583 


3776 


3968 4161 


4353 


4546 


4738 


4931 


5123 


192 


7 


5508 


5700 


5893 6085 


6278 


6470 


6662 


6855 


7047 


2 


8 


7432 


7624 


7816 8009 


8201 


8393 


8586 


8778 


8970 


2 


9 


9355 


9547 


9739 9931 


3540123 


0316 


3540508 


0700 


3540892 


2 


2260 


3541277 


1469 


3541661 1853 


2045 


2237 


2429 


2621 


2814 


2 


1 


3198 


3390 


3582 3774 


3966 


4158 


4350 


4542 


4734 


2 


2 


5118 


5310 


5502 5694 


5886 


6078 


6270 


6462 


6654 


2 


3 


7037 


7229 


7421 7613 


7805 


7997 


8189 


8381 


8572 


2 


4 


8956 


9148 


9340 9531 


9723 


9915 


3550107 


0299 


3550490 2 || 


5 


3550874 


1066 


3551257 1449 


3551641 


1832 


2024 


2216 


2407 


2 


6 


2791 


2982 


3174 3366 


3557 


3749 


3940 


4132 


4324 


2 


7 


4707 


4898 


5090 5281 


5473 


5664 


5856 


6048 


6239 


2 


8 


6622 


6813 


7005 7196 


7388 


7579 


7771 


7962 


8154 


191 


9 


8536 


8728 


8919 9111 


9302 


9493 


9685 


9876 


3560067 




2270 


3560450 


0641 


3560832 1024 


3561215 


1406 


3561598 


1789 


1980 




1 


2363 


2554 


2745 2936 


3127 


3319 


3510 


3701 


3892 




2 


4274 


4466 


4657 4848 


5039 


5230 


5421 


5612 


5803 




3 


6185 


6376 


6568 6759 


6950 


7141 


7332 


7523 


7714 




4 


8096 


8287 


8478 8668 


8859 


9050 


9241 


9432 


9623 




5 


3570005 


0196 


3570387 0578 


3570768 


0959 


3571150 


1341 


3571532 




6 


1913 


2104 


2295 2486 


2677 


2867 


3058 


3249 


3440 




7 


3821 


4012 


4202 4393 


4584 


4775 


4965 


5156 


5347 




8 


5728 


5918 


6109 6300 


6490 


6681 


6872 


7062 


7253 




9 


7634 


7824 


8015 8205 


8396 


85861 8777 
6 1 


8967 


9158 






1 


2 


3 4 


5 


8 


9 


= 



40 


LOGARITHMS 


3F NUMBERS FROM 1 TO 36,000. {TobU I. 


Between 22800 = log." 


1 4-3579348, and 23400 = log.-i 4-3692159. 


tens. 


1 


2 


3 


^ 1 


5 


6 


T 8 


9 


dif. 


2280 


3579539 


9729 


3579920/01101 


3580301 


0491 


3580682 0872 


3581062 


190 


1 


3581443 


1634 


3581824^ 


2014 


2205 


2395 


2585 2776 


2966 





2 


3347 


3537 


3727 


3918 


4108 


4298 


4488 4679 


4869 





3 


5249 


5440 


5630 


5820 


6010 


6200 


6391 6581 


6771 





4 


7151 


7341 


7531 


7722 


7912 


8102 


8292 8482 


8672 





5 


9052 


9242 


9432 


9622 


9812/00021 


3590192 0382 


3590572 





6 


3590952 


1142 


3591332 


1522 


3591712 ' 


1902 


2092 2282 


2472 





7 


2852 


3041 


3231 


3421 


3611 


3801 


3991 4181 


4370 





8 


4750 


4940 


5130 


5319 


5509 


5699 


5889 6078 


6268 





9 


6648 


6837 


7027 


7217 


7406 


7596 


7786 7976 


8165 





2290 


8544 


8734 


8924 


9113 


9303 


9493 


9682 9872 


3600061 





1 


3600440 


0630 


3600820 


1009 


3601199 


1388 


3601578 1767 


1957 





2 


2336 


2525 


2715 


2904 


3093 


3283 


3472 3662 


3851 


189 


3 


4230 


4419 


4609 


4798 


4987 


5177 


5366 5555 


5745 


9 


4 


6123 


6313 


6502 


6691 


6881 


7070 7259 7448 


7638 


9 


5 


8016 


8205 


8395 


8584 


8773 


8962 


9151 9341 


9530 


9 


6 


9908 / 
3611799 ' 


0097 


3610286 


0475 


3610664 


0854 


3611043 1232 


3611421 


9 


7 


1988 


2177 


2366 


2555 


2744 


2933 3122 


3311 


9 


8 


3689 


3878 


4067 


4256 


4445 


4634 


4823 5012 


5201 


9 


9 


5579 


5768 


5956 


6145 


6334 


6523 


6712 6901 


7090 


9 


2300 


7467 


7656 


7845 


8034 


8222 


8411 


8600 8789 


8977 


9 


1 


9355 


9544 


9732 


9921 


3620110 


0298 


3620487 0676 


3620865 


9 


2 


3621242 


1430 


3621619 


1808 


1996 


2185 


2374 2562 


2751 


9 


3 


3128 


3317 


3505 


3694 


3882 


4071 


4259 4448 


4636 


9 


4 


5013 


5202 


5390 


5579 


5767 


5956 


6144 6332 


6521 


188 


5 


6898 


7086 


7275 


7463 


7651 


7840 


8028 8216 


8405 


8 


6 


8781 


8970 


9158 


9346 


9535 


9723 


9911 /0099 
3631794 ' 1982 


3630238 


8 


7 


3630664 


0852 


3631041 


1229 


3631417 


1605 


2170 


8 


8 


2546 


2734 


2923 


3111 


3299 


3487 


3675 3863 


4051 


8 


9 


4427 


4615 


4804 


4992 


5180 


5368 


5556 5744 


5932 


8 


2310 


6308 


6496 


6684 


6872 


7060 


7248 


7436 7624 


7812 


8 


1 


8187 


8375 


8563 


8751 


8939 


9127 


9315 9503 


9690 


8 


2 


3640066 


0254 


3640442 


0630 


3640817 


1005 


3641193 1381 


3641569 


8 


3 


1944 


2132 


2320 


2507 


2695 


2883 


3070 3258 


3446 


8 


4 


3821 


4009 


4197 


4384 


4572 


4759 


4947 5135 


• 5322 


8 


5 


5698 


5885 


6073 


6260 


6448 


6635 


6823 7010 


7198 


8 


6 


7573 


7761 


7948 


8136 


8323 


8511 


8698 8885 


9073 


187 


7 


9448 


9635 


9823 


/OOIO 

'l884 


3650197 


0385 


3650572 0760 


3650947 




8 


3651322 


1509 


3651696 


2071 


2258 


2446 2633 


2820 




9 


3195 


3382 


3569 


3757 


3944 


4131 


4318 4505 


4693 




2320 


5067 


5254 


5441 


5629 


5816 


6003 


6190 6377 


6564 




1 


6939 


7126 


7313 


7500 


7687 


7874 


8061 8248 


8435 




2 


8809 


8996 


9183 


9370 


9557 


9744 


9931/0118 
3661801 ' 1987 


3660305 




3 


3660679 


0866 


3661053 


1240 


3661427 


1614 


2174 




4 


2548 


2735 


2922 


3109 


3296 


3482 


3669 3856 


4043 




5 


4416 


4603 


4790 


4977 


5163 


5350 


5537 5724 


5910 




6 


6284 


6471 


6657 


6844 


7031 


7217 


7404 7591 


7777 




7 


8150 


8337 


8524 


8710 


8897 


9083 


9270 9457 


9643 




8 


3670016 


0203 


3670389 


0576 


3670762 


0949 


3671135 1322 


3671508 




9 


1881 


2068 


2254 


2441 


2627 


2814 


3000 3186 


3373 


186 


2330 


3746 


3932 


4118 


4305 


4491 


4677 


4864 5050 


5236 


6 


1 


5609 


5795 


5982 


6168 


6354 


6540 


6727 6913 


7099 


6 


2 


7472 


7658 


7844 


8030 


8217 


8403 


8589 8775 


8961 


6 


3 


9334 


9520 


9706 


9892 


3680078 


0264 


3680450 0636 


3680822 


6 


4 


3681195 


1381 


3681567 


1753 


1939 


2125 


2311 2497 


2683 


6 


5 


3055 


3241 


3427 


3613 


3799 


3985 


4171 4357 


4542 


6 


6 


4914 


5100 


5286 


5472 


5658 


5844 


6030 6215 


6401 


6 


7 


6773 


6959 


7145 


7330 


7516 


7702 


7888 8074 


8259 


6 


e 


8631 


8817 


9002 


9188 


9374 


9559 


9745 9931 


3690117 


6 


c 


3690488 


0674 


3690859 


1045 


3691230 


1416 


3691602 1787 


1973 


6 


1 1 


2 1 3 


4 


5 


6 


T 8 


9 


- ' 



Table I.] 


LOGARITHMS 


OF NUMBERS 


FROM 1 TO 36,000. 41 ll 


Between 23400 rr: log. - 


■1 4-3692159, and 2400O = log.-i 4-38021 12. 11 


tens. 


I i 


g 


3 


4 \ 5 


6 T 


8 


9 \dif. II 


2340 


3692344 


2530 


3692715 


2901 


3693086 


3272 3693458 


3643 


3693829 


186 


1 


4200 


4385 


4571 


4756 


4942 


5127 5313 


5498 


5683 


5 


2 


6054 


6240 


6425 


6611 


6796 


6981 7167 


7352 


7538 


5 


3 


790S 


8094 


8279 


8464 


8650 


8835 9020 


9205 


9391 


5 


4 


9761 


9947 


3700132 


0317 


3700502 


0688 3700873 


1058;3701243 


5 ! 


5 


3701614 


1799 


1984 


2169 


2354 


2540 


2725 


2910 


3095 


5 


6 


3456 


3650 


3835 


4020 


4206 


4391 


4576 


4761 


4946 


5 


7 


5316 


5501 


5686 


5871 


6056 


6241 


6426 


6611 


6796 


5 


8 


7166 


7351 


7536 


7721 


7906 


8091 


8275 


8460 


8645 


5 


9 


9015 


920O 


9385 


9570 


9754 


9939 


3710124 


0309 


3710494 


5 


2350 


3710863 


1048 


3711233 


1418 


3711603 


1787 


1972 


2157 


2342 


5 


1 


2711 


2896 


3080 


3265 


3450 


3635 


3819 


4004 


4189 


5 


2 


4558 


4742 


4927 


5112 


5296 


5481 


5666 


5850 


6035 


5 


3 


6404 


6588 


6773 


6957 


7142 


7327J 7511 
9171' 9356 


7696 


7880 


5 


4 


8249 


8434 


8618 


8802 


8987 


9540' 9725 


184 


5 


3720094 


0278 


3720462 


0647 


3720831 


1015 


3721200 


1384 


3721569 


4 


6 


1937 


2122 


2306 


2490 


2674 


2859 


3043 


3227 


3412 


4 


7 


3780 


3964 


4149 


4333 


4517 


4701 


4885 


5070 


5254 


4 


8 


5622 


5806 


5991 


6175 


6359 


6543 


6727 


6911 


7095 


4 


9 


7464 


7648 


7832 


8016 


8200 


8384 


8568 


8752 


8936 


4 


2360 


9304 


9488 


9672 


9856 


3730040 


0224 


3730408 


0592 


3730776 


4 


1 


3731144 


1328 


3731512 


1696 


1879 


2063 


2247 


2431 


2615 


4 


2 


2983 


3167 


3350 


3534 


3718 


3902 


4086 


4270 


4453 


4 


3 


4821 


5005 


5189 


5372 


5556 


5740 


5924 


6107 


6291 


4 


4 


6658 


6842 


7026 


7210 


7393 


7577 


7761 


7944' 8128 


4 


5 


8495 


8679 


8862 


9046 


9230 


9413 


9597 


9780 


9964 


4 


6 


3740331 


0515 


3740698 


0882 


3741055 


1249 


3741432 


1616 


3741799 


4 


7 


2166 


2350 


2533 


2716 


2900 


3083 


3267 


3450 


, 3634 


183 


8 


4000 


4184 


4367 


4551 


4734 


4917 


5101 


5284 


5467 


3 


9 


5834 


6017 


6201 


6384 


6567 


6750 


6934 


7117 


7300 


3 


2370 


7667 


7850 


8033 


8216 


8400 


8583 


8766 


8949 


9132 


3 


1 


9499 


9682 


9865/ 
3751696 ' 


0048 


3750231 


0414 


3750598 


0781 


3750964 


3 


2 


3751330 


1513 


1879 


2062 


2245 


2428 


2611 


2794 


3 


3 


3160 


3343 


3526 


3709 


3892 


4075 


4258 


4441 


4624 


3 


4 


4990 


5173 5356 


5539 5722 


5905 


6088 


6270 


6453 


3 


5 


6819 


7002 


7185 


7367 


7550 


7733 


7916 


8099 


8282 


3 


6 


8647 


8830 


9013 


9195 


9378 


9561 


9744 


9926 


3760109 


3 


7 


3760475 


0657 


3760840 


1023 


3761205 


1388 


3761571 


1753 


1936 


3 


8 


2301 


2484 


2666 


2849 


3032 


3214 


3397 


3579 


3762 


3 


9 


4127 


4310 


4492 


4675 


4857 


5040 


5222 


5405 


5587 


3 


2380 


5952 


6135 


6317 


6499 


6682 


6864 


7047 


7229 


7412 


182 


1 


7776 


7959 


8141 


8323 


8506 


8688 


8871 


9053 


9235 


2 


2 


9600 


9782 


9965 


,0147 


3770329 


0511 


3770694 


0876 


3771058 


2 


3 3771423 

4 3245 


1605 


3771787 / 1969 


2152 


2334 


2516 


2698 


2880 


2 


3427 


3609 


3791 


3973 


4155 


4338 


4520 


4702 


2 


5 


5066 


5248 


5430 


5612 


5794 


5976 


6158 


6340 


6522 


2 


6 


6886 


7068 


7250 


7432 


7614 


7796 


7978 


8160 


8342 


2 


7 


8706 


8888 


9070 


9252 


9434 


9616 


9798 


9979 


3780161 


2 


8 


3780525 


0707 


3780889 


1071 


3781252 


1434 


3781616 


1798 


1980 


2 


9 


2343 


2525 


2707 


2689 


3070 


3252 


3434 


3616 


3797 


2 


2390 


4161 


4342 


4524 


4706 


4887 


5069 


5251 


5432 


6614 


2 


1 


5977 


6159 


6341 


6522 


6704 


6885 


7067 


7249 


7430 


2 


2 


7793 


7975 


8156 


8338 


8519 


8701 


8882 


9064 


9245 


2 


3 


9608 


9790 


9971/ 
3791786^ 


0153 


3790334 


0516 


3790697 


0879 


3791060 


181. 


4 


3791423 


1604 


1967 


2148 


2330 


2511 


2692 


2874 




5 3237 


3418 3599 


3780 


3962 


4143 


4324 


4506 


4687 




6 5049 


5231 5412 


5593 


5774 


5956 


6137 


6318 


6499 




7 6862 


7043 7224 


7405 


7586 


7767 


7948 


8130 


8311 




8 8673 


8854 9035 


9216 


9397 


9578 


9759 


9940 


3800121 




9 3S00484 


0665 3800846 


1027 


3801208 


1389 


3801570 


1750 


1931 


'■ 


I . ^ 


2 3 


4 


5 


6 


T 


8 9 1 


J 



4* 



42 LOGARITHMS OF NUMBERS FROM 1 TO 36,000. [Table I. {| 


Between 24000 = log." 


1 4-3802112, and 24600 = log "^ 4-3909351. 1 


tens. 


1 


2 


3 


* 


5 


6 


T 8 


9 


dif. 1 


2400 


3802293 


2474 


3802655 


2836 


3803017 


3198 


3803379 3560 


3603741 


181 


1 


4102 


4283 


4464 


4645 


4826 


5007 


5188 5368 


5549 




2 


5911 


6092 


6272 


6453 


6634 


6815 


6995 7176 


7357 




3 


7718 


7899 


8080 


8261 


8441 


8622 


8803 8983 


9164 




4 


9525 


9706 


9887/ 
3811693' 


0067 3810248 


0428 


3810609 0790 3810970' 




5 


3811331 


1512 


1873 


2054 


2234 


2415 2595 


2776 




6 


3137 


3317 


3498 


3678 


3859 


4039 


4220 4400 


4580 





7 


4941 


5122 


5302 


5483 


5663 


5843 


6024 6204 


6384 





8 


6745 


6926 


7106 


7286 


7467 


7647 


7827 8007 


8188 





9 


8548 


8729 


8909 


9089 


9269 


9450 


9630 9810 


9990 





2410 


3820351 


0531 


3820711 


0891 


3821071 


1252 


3821432 1612 


3821792 





1 


2152 


2332 


2512 


2693 


2873 


3053 


3233 3413 


3593 





2 


3953 


4133 


4313 


4493 


4673 


4853 


5033 5213 


5393 





3 


5753 


5933 


6113 


6293 


6473 


6653 


6833 7013 


7193 





4 


7553 


7732 


7912 


8092 


8272 


8452 8632 8812' 8992 





5 


9351 


9531 


9711 


9891 


3830070 


0250 


3830430 0610 


3830790 





6 


3831149 


1329 


3831509 


1688 


1868 


2048 


2227 2407 


2587 





7 


2946 


3126 


3306 


3485 


3665 


3844 


4024 4204 


4383 





8 


4743 


4922 


5102 


5281 


5461 


5640 


5820 6000 


6179 





9 


6538 


6718 


6897 


7077 


7256 


7436 


7615 7795 


7974 


179 


2420 


8333 


8513 


8692 


8871 


9051 


9230 


9410 9589 


9769 


9 


1 


3840127 


0307 


3840486 


0665 


3840845 


1024 


3841203 1383 


3841562 


9 


2 


1921 


2100 


2279 


2459 


2638 


2817 


2996 3176 


3355 


9 
9 


3 


3713 


3893 


4072 


4251 


4430 


4609 


4789 4968 


5147 


4 


5505 


5684 


5864 


6043 


6222 


6401 


6580 6759^ 6938 


9 


5 


7297 


7476 


7655 


7834 


8013 


8192 


8371 8550 


8729 


9 


6 


9087 


9266 


9445 


9624 


9803 


9982 


3850161 0340 


38505 IS 


9 


7 


3850877 


1056 


3851235 


1413 


3851592 


1771 


1950 2129 


2308 


9 


8 


2666 


2845 


3023 


3202 


3381 


3560 


3739 3918 


4096 


9 


9 


4454 


4633 


4812 


4990 


5169 


5348 


5527 5705 


5884 


9 


2430 


6241 


6420 


6599 


6778 


6956 


7135 


7314 7492 


7671 


9 


1 


8028 


8207 


8386 


8564 


8743 


8921 


9100 9279 


9457 


9 


2 


9814 


9993 


3860171 


0350 


3860528 


0707 


3860886 1064 


3861243 


9 


3 


3861600 


1778 


1957 


2135 


2314 


2492 


2670 2849 


3027 


178 


4 


3384 


3563 


3741 


3919 


4098 


4276 4455 4633 


4811 8 II 


5 


5168 


5346 


5525 


5703 


5881 


6060 


6238 6416 


6595 


8 


6 


6951 


7129 


7308 


7486 


7664 


7842 


8021 8199 


8377 


8 


7 


8733 


8912 


9090 


9268 


9446 


9624 


9803 9981 


3870159 


8 


8 


3870515 


0693 


3870871 


1049 


3871228 


1406 


3871584 1762 


1940 


8 


9 


2296 


2474 


2652 


2830 


3008 


3186 


3364 3542 


3720 


8 


2440 


4076 


4254 


4432 


4610 


4788 


4966 


5144 5322 


5500 


8 


1 


5856 


6034 


6212 


6389 


6567 


6745 


6923 7101 


7279 


8 


2 


7934 


7812 


7990 


8168 


8346 


8524 


8701 8879 


9057 


8 


3 


9412 


9590 


9768 


9946 


3880123 


0301 


3880479 0657 


3880834 


8 


4 


3881190 


1367 


3881545 


1723 


1900 


2078 


2256 2433 


2611 


8 


5 


2966 


3144 


3321 


3499 


3677 


3854 


4032 4209 


4387 


8 


6 


4742 


4920 


5097 


5275 


5452 


5630 


5807 5985 


6162 


8 


7 


6517 


6695 


6872 


7050 


7227 


7404 


7582 7759 


7937 


177 


8 


8292 


8469 


8646 


8824 


9001 


9178 


9356 9533 


9711 


7 


9 


3890065 


0243 


3890420 


0597 


3890774 


0952 


3891129 1306 


3891484 


7 


2450 


1838 


2015 


2193 


2370 


2547 


2724 


2902 3079 


3256 


7 


1 


3610 


3787 


3965 


4142 


4319 


4496 


4673 4850 


5028 


7 


2 


5382 


5559 


5736 


5913 


6090 


6267 


6444 6621 


6798 


7 


3 


7153 


7330 


7507 


7684 


7861 


8038 


8215 8392 


8569 


7 


4 


8923 


9100 


9276 


9453 


9630 


9807 


9984 , 0161 


3900338 


7 


5 


3900692 


0869 


3901046 


1223 


3901399 


1576 


3901753/1930 


2107 


7 


6 


2460 


2637 


2814 


2991 


3168 


3344 


3521 3698 


3875 


7 


7 


4228 


4405 


4582 


4759 


4935 


5112 


5289 5465 


5642 


7 


8 


5995 


6172 


6349 


6525 


6702 


6879 


7055 7232 


7409 


7 


9 


7762 


7939 


8115 


8292 


8468 


8645 


8821 8998 


9175 


7 


■ 


1 


3 


3 


4 


5 


« 


1 8 1 9 





Table I.] 


LOGARITHMS OF NUMBERS 


FROM 1 TO 36,000. 43 ll 


Between 24600 = log "i 4-3909351, and 25200 = log.-i 4-4014005. 11 


tejis. 
2460 


1 


2 3 4 1 


5 


6 


T 


8 


9 


dif. 


3909528 


9704 


3909881 /0057 
3911646^ 1822 


3910234 


0410 


3910587 


0763 


3910940 


177 


1 


3911293 


1469 


1998 


2175 


2351 


2528 


2704 


6 


2 


3057 


3233 


3410 3586 


3762 


3939 


4115 


4291 


4468 


6 


3 


4820 


4997 


5173 5349 


5526 


5702 


5878 


6055 


6231 


6 


4 


6583 


6760 


6936 7112 


7288 


7464 


7641 


7817 


7993 


6 


5 


8345 


8522 


8698 8874 


9050 


9226 


9402 


9578 


9755 


6 


6 


3920107 


0283 


3920459 0635 


3920811 


0987 


3921163 


1339 


3921515 


6 


7 


1868 


2044 


2220 2396 


2572 


2748 


2924 


3100 


3276 


6 


8 


3628 


3803 


3979 4155 


4331 


4507 


4683 


4859 


5035 


6 


9 


5387 


5563 


5739 5914 


6090 


6266 


6442 


6618 


6794 


6 


2470 


7145 


7321 


7497 7673 


7849 


8024 


8200 


8376 


8552 


6 


1 


8903 


9079 


9255 9430 


9606 


9782 


9958 


.0133 


3930309 


6 


2 


3930660 


0836 


3931012 1187 


3931363 


1539 


3931714 / 1890 


2066 


6 


3 


2417 


2592 


2768 2944 


3119 


3295 


3470 


3646 


3821 


6 


4 


4172 


4348 


4524 4699 


4875 


5050 


5226 


5401 


5577 


6 


5 


5928 


6103 


6278 6454 


6629 


6805 


6980 


7156 


7331 


175 


6 


7682 


7857 


8033 8208 


8383 


8559 


8734 


8909 


9085 


5 


7 


9435 


9611 


9786 9961 


3940137 


0312 


3940487 


0662 


3940838 


5 


8 


3941188 


1364 


3941539" 1714 


1889 


2064 


2240 


2415 


2590 


5 


9 


2940 


3116 


3291 3466 


3641 


3816 


3991 


4167 


4342 


5 


2480 


4692 


4867 


5042 5217 


5392 


5567 


5742 


5918 


6093 


5 


1 


6443 


6618 


6793 6968 


7143 


7318 


7493 


7668 


7843 


5 


2 


8193 


8368 


8543 8718 


8893 


9068 


9242 


9417 


9592 


5 


3 


9942/01171 


3950292 0467 


3950642 


0817 


3950991 


1166 


3951341 


5 


4 


3951691 ' 


1866' 


2040 2215 


2390 


2565 


2740 


2914 


3089 


5 


5 


3439 


3613 


3788 3963 


4138 


4312 


4487 


4662 


4837 


5 


6 


5186 


5361 


5535 5710 


5885 


6059 


6234 


6409 


6583 


5 


7 


6932 


7107 


7282 7456 


7631 


7805 


7980 


8155 


8329 


5 


8 


8678 


8853 


9027 9202 


9376 


9551 


9725 


9900 


3960074 


5 


9 


3960423 


0598 


3960772 0947 


3961121 


1296 


3961470 


1645 


1819 


174 


2490 


2168 


2342 


2517 2691 


2865 


3040 


3214 


3389 


3563 




1 


3912 


4086 


4260 4435 


4609 


4783 


4953 


5132 


5306 




2 


5655 


5829 


6003 6177 


6352 


6526 


6700 


6874 


7049 




3 


7397 


7571 


7745 7920 


8094 


6268 


8442 


8616 


8790 




4 


9139 


9313 


9487 9661 9835 


,0009 
M750 


3970183 


0357 


3970531 




5 


3970880 


1054 


3971228 1402 


3971576 


1924 


2098 


2272 




6 


2620 


2794 


2968 3142 


3316 


3490 


3664 


3H38 


4011 




7 


4359 


4533 


4707 4881 


5055 


5229 


5403 


5577 


5750 




8 


6098 


6272 


6446 6620 


6794 


6967 


7141 


7315 


7489 




9 


7836 


8010 


8184 8358 


8531 


8705 


8879 


9053 


9226 




2500 


9574 


9748 


9921 ,0095 
3981658/1831 


3980269 


0442 


3980616 


0790 


3980963 




1 


3981311 


1484 


2005 


2179 


2352 


2526 


2699 




2 


3047 


3220 


3394 3567 


3741 


3914 


4088 


4261 


4435 




3 


4782 


4956 


5129 5302 


5476 


5649 


5823 


5996 


6170 


173 


4 


6517 


6690 


6864 7037 


7210 


7384 


7557 


7731 


7904 




5 


8251 


8424 


8597 8771 


8944 


9117 


9291 


9464 


9637 


3 


6 


9984 


,0157 
'1890 


3990331 0504 


3990677 


0850 


3991024 


1197 


3991370 


3 


7 


3991717 


2063 2236 


2409 


2583 


2756 


2929 


3102 


3 


8 


3448 


3622 


3795 3968 


4141 


4314 


4487 


4660 


4834 


3 


9 


5180 


5353 


5526 5699 


5872 


6045 


6218 


6391 


6564 


3 


2510 


6910 


7083 


7256 7429 


7602 


7775 


7948 


8121 


8294 


3 


1 


8640 


8813 


8986 9159 


9332 


9505 


9678 


9851 


4000023 


3 


2 


4000369 


0542 


4000715 0888 


4001061 


1234 


4001406 


1579 


1752 


3 


3 


2098 


2271 


2443 2616 


2789 


2962 


3134 


3307 


3480 


3 


4 


3825 


3998 


4171 4344 


4516 


4689 


4862 


5035 


5207 


3 


5 


5553 


5725 


5898 6071 


6243 


6416 


6588 


6761 


6934 


3 


6 


7279 


7452 


7624 7797 


7969 


8142 


8314 


8487 


8660 


3 


7 


9005 


9177 


9350 9522 


9695 


9867 


4010040 


0212 


4010385 


3 


8 


4010730 


0902 


4011075 1247 


4011420 


1592 


1764 


1937 


2109 


172 


9 


2454 


2626 


2799 2971 


3144 


3316 


3488 


3661 


3833 


2 




1 


2 


3 4 


5 


6 


T 


8 


9 





44 


LOGARITHMS OF NUMBERS FROM 1 TO 36,000. [Table I. 1 


Between 25200 = log." 


' 4-4014005, arid 25800 = log "^ 4-4116197. j 


tens. 


1 


3 


3 


4: 5 


6 


T 8 


9 


dif. 


2520 


4014178 


4350 


4014522 


4695 4014867 


5039 


4015212 5384 


4015556 


172 


1 


5901 


6073 


6245 


6417 6590 


6762 


6934 7106 


7279 


2 


2 


7623 


7795 


7967 


8140 8312 


8484 


8656 8828 


9000 


2 


3 


9345 


9517 


9689 


9361 4020033 


0-205 


4020377 0549 


4020721 


2 


4 


4021066 


1238^4021410 


1582^ 1754 


1926° 2098 2270* 2442 » 


2 


1 5 2786 


2958 


3130 


3302 3474 


3646 


3818 3990 


4162 


2 


6 4505 


4677 


4849 


5021 5193 


5365 


5537 5709 


5881 


2 


7 6224 


6396 


6568 


6740 6912 


7033 


7255 7427 


7599 


2 


8 7942 


8114 


8286 


8458 8630 


8801 


8973 9145 


9317 


2 


9 


9660 


9832 


4030003 


0175 4030347 


0519 


4030690 0862 


4031034 


2 


2530 


4031377 


1549 


1720 


1892 2063 


2235 


2407 2578 


2750 


2 


1 


3093 


3265 


3436 


3608 3779 


3951 


4122, 4294 


4465 


2 


2 


4809 


4980 


5152 


5323 5495 


5666 


5838 6009 


6180 


171 


3 


6523 


6695 


6366 


7038 7209 


7331 


7552 7723 


7895 




4 


8237 


8409 


8580 


8752' 8923 


9094 


9266 9437 


9608 




5 


9951 


,0122 


4040294 


0465 


4040636 


0307 


4040979 1150 
2691 2862 


4041321 




6 


4041664/ 1835 


2006 


2177 


2349 


2520 


3033 




7 


3376 


3547 


3713 


3839 


4061 


4232 


4403 4574 


4745 




8 


5087 


5258 


5429 


5601 


5772 


5943 


6114 6285 


6456 




9 


6798 


6969 


7140 


7311 


7482 


7653 


7824 7995 


8166 




2540 


8508 


8679 


8850 


9021 


9192 

4050901 


9363 


9534 9705 


9876 




1 


4050218 


0388 


4050559 


0730 


1072 


4051243 1414 


4051585 




2 


1926 


2097 


2268 


2439 


2610 


2780 


2951 3122 


3293 




3 


2634 


3805 


3976 


4147 


4317 


4488 


4659 4830 


5000 




4 


5342 


5512 


5683 


5354 


6025 


6195* 6366 6537 


6707 




5 


7049 


7219 


7390 


7560 


7731 


7902 


8072 8243 


8413 




6 


8755 


8925 


9096 


9266 


9437 


9607 


9778 9948 


4060119 




7 


4060460 


0630 


4060801 


0971 


4061142 


1312 


4061483 1653 


1824 


170 


8 


2165 


2335 


2506 


2676 


2846 


3017 


3187 3358 


3528 





9 


3869 


4039 


4209 


4380 


4550 


4721 


4891 5061 


5231 





2550 


5572 


5742 


5913 


6083 


6253 


6424 


6594 6764 


6934 





1 


7275 


7445 


7615 


7785 


7956 


8r26 


8296 8466 


8637 





2 


8977 


9147 


9317 


9487 


9658 


9828 


9998 / 0168 
4071699 ' 1869 


4070338 





3 


1070678 


0848 


4071018 


1189 


4071359 


1529 


2039 





4 


2379 


2549 


2719 


2889 


3059 


3229 


3399 3569 


3739 





5 


4079 


4249 


4419 


4589 


4759 


4929 


5099 5-269 


5439 





6 


5778 


5948 


6118 


6288 


6458 


6628 


6798 6968 


7137 





7 


7477 


7647 


7817 


7987 


8156 


8326 


8496 8666 


8836 





8 


9175 


9345 


9515 


9684 


9854 / 0024 


4080194 0363 


4080533 





9 


4080873 


1042 


4081212 


1382 


4081551' 


1721 


1891 2060 


2230 





2560 


2569 


2739 


2909 


3078 


3248 


3417 


3587 3757 


3926 





1 


4265 


4435 


4604 


4774 


4944 


5113 


5283 5452 


5622 


169 


2 


5961 


6130 


6300 


6469 


6639 


6808 


6978 7147 


7317 


9 


3 


7656 


7825 


7994 


8164 


8333 


8503 


8672 8841 


9011 


9 


4 


9350 


9519 


9688 


9858 


4090027 


0196 


4090366 0535 


4090704 


9 


5 


4091043 


1212 


4091382 


1551 


1720 


1889 


2059 2228 


2397 


9 


6 


2736 


2905 


3074 


3243 


3413 


3582 


3751 3920 


4089 


9 


7 


4428 


4597 


4766 


4935 


5105 


5274 


5443 5612 


5781 


9 


8 


6119 


6288 


6458 


6627 


6796 


6965 


7134 7303 


7472 


9 


9 


7810 


7979 


8148 


8317 


8486 


8655 


8824 8993 


9162 


9 


2570 


9500 


9669 


9838/ 
4101527 ' 


0007 


4100176 


0345 


4100514 0683 


4100852 


9 


1 


4101190 


1359 


1696 


1865 


2034 


2203 2372 


2541 


9 


2 


2878 


3047 


3216 


3385 


3554 


3723 


3891 4060 


4229 


9 


3 


4567 


4735 


4904 


5073 


5242 


5410 


5579 5748 


5917 


9 


4 


6254 


6423 


6592 


6760 


6929 


7098 


7266' 7435 


7604 


9 


5 


7941 


8110 


8273 


8447 


8616 


8784 


8953 9121 


9290 


9 


6 


-9627 


9796 


9984/ 
4111650' 


0133 


4110301 


0470 


4110639 0807 


4110976 


9 


7 


4111313 


1481 


1818 


19S7 


2155 


2324 2492 


2661 


168 


8 


2998 


3166 


3334 


3503 


3671 


3840 


4003 4177 


4345 


8 


9 


4682 


4850 


5019 


5187 


5355 


5524 


5692 5860 


6029 


8 




1 


2 


3 


4 § 


6 


T 8 


9 





Table i.] logarithms of numbers from 1 to 36,000. 45 j| 


Between 25800 = log.'" 


' 4-4116197, and 26400 ..= log.-i 4-4216039. 1 


tens. 1 


2 

6534 


3 


^ 


o 


1 


'^ ® 1 


«» 


dif. 


2530 


411G365 


4116702 


6870 


4117039 


7207 


4117375 7544 


4117712 


168 


1 


8048 


8217 


8385 


8553 


8721 


8890 


9058 9226 


9394 


8 


2 


9731 


9899 


4120067 


0235 


4120403 


0571 


4120740 0908 


4121076 


8 


3 


4121412 


1580 


1748 


1917 


2085 


2253 


2421 2589 


2757 


8 


4 


3093 


3261 


3429 


3597 


3765 


3933 


4101 4269 4437 


8 


5 


4773 


4941 


5109 


5277 5445 


5613 


5781 5949 


6117 


8 


6 


6453 


6621 


6789 


6957 7125 


7293 


7461 7629 


7796 


8 


7 


8132 


8300 


8468 


8636 8804 


8971 


9139 9307 


9475 


8 


8 


9811 


9978 


4130146 


0314 


4130482 


0649 


4130617 0985 


4131153 


8 


9 


4131488 


1656 


1824 


1991 


2159 


2327 


2495 2662 


2830 


8 


2590 


3165 


3333 


3501 


3668 


3836 


4004 


4171 4339 


4507 


8 


1 


4842 


5009 


5177 


5345 


5512 


5680 


5847 6015 


6182 


8 


2 


6518 


6685 


6853 


7020 


7183 


7355 


7523 7690 


7858 


8 


3 


8193 


8360 


8528 


8695 


8863 


9030 


9197 9365 


9532 


167 


4 


9867 


/0035 
1708 


4140202 


0369^4140537 


0704 


4140872 1039M141206 


7 


5 


4141541 


1876 


2043 


2210 


2378 


2545 2712 


2880 


7 


6 


3214 


3381 


3549 


3716 


3883 


4051 


4218 4385 


4552 


7 


7 


4887 


5054 


5221 


5388 


5556 


5723 


5890 6057 


0224 


7 


8 


6559 


6726 


6893 


7060 


7227 


7394 


7561 7729 


7896 


7 


9 


8230 


8397 


8564 


8731 8898 


9065 


9232 9399 


9566 


7 


2600 


9901 


/006S 
1737 


4150235 


0402 4150569 


0736 


4150903 1070 


4151237 


7 


1 


4151570 


1904 


2071 2238 


2405 


2572 2739 


2906 


7 


2 


3240 


3407 


3574 


3741 3907 


4074 


4241 4408 


4575 


7 


3 


4909 


5075 


5242 


5409 5576 


5743 


5909 6076 


6243 


7 


4 


6577 


6743 


6910 


7077 7244 


7410 


7577 7744^ 7911 


7 


5 


8244 


84111 


8577 


8744 


8911 


9077 


9244 9411 


9577 


7 


6 


9911 /0077J 


4160244 


0411 


4160577 


0744 


4160911 1077 


4161244 


7 


7 


4161577' 


1743 


1910 


2077 


2243 


2410 


2576 2743 


2909 


7 


8 


3242 


3409 


3575 


3742 


3908 


4075 


4241 4408 


4574 


166 


9 


4907 


5074 


5240 


5407 


5573 


5739 


5906 6072 


6239 


6 


2610 


6571 


6738 


6904 


7071 


7237 


7403 


7570 7736 


7902 


6 


1 


8235 


8401 


8568 


8734 


8900 


9067 


9233 9399 


9565 


6 


2 


9898 / 00641 


4170231 


0397 


4170563 


0729 


4170895 1062 


4171228 
2890 


6 


3 


4171560 ' 


1726 


1893 


2059 


2225 


2391 


2557 2724 


6 


4 


3222 


3388 


3554 


3720 


3886 


4053 


4219 4385'' 4551" 6 !| 


5 


4883 


5049 


5215 


5318 


5547 


5713 


5879 6045 


6211 


6 


6 


6543 


6709 


6875 


7041 


7207 


7373 


7539 7705 


7871 


6 


7 


8203 


8369 


8535 


8701 


8867 


9033 


9199 9365 


9531 


6 


8 


9862 


/0028 
1687 


4180194 


0360 


4180526 


0692 


4180857 1023 


4181189 


6 


9 


4181521 ^ 


1852 


2018 


2184 


2350 


2516 2681 


2847 


6 


2620 


3179 


3344 


3510 


3676 


3842 


4007 


4173 4339 


4505 


6 


1 


4836 


5002 


5167 


5333 


5499 


5664 


5830 5996 


6161 


6 


2 


6493 


6658 


6824 


6989 


7155 


7321 


7486 7652 


7817 


6 


3 


8148 


8314 


8480 


8645 


8811 


8976 


9142 9307 


9473 


6 


4 


9804 


9969 


4190135 


0300 


4190466 


0631 


4190797 0962 


4191128 


165 


5 


4191459 


1624 


1789 


1955 


2120 


2286 


2451 2616 


2782 


5 


6 


3113 


3278 


3443 


3609 


3774 


3939 


4105 4270 


4435 


5 


7 


4766 


4931 


5097 


5262 


5427 


5593 


5758 5923 


6088 


5 


8 


6419 


6584 


6749 


6915 


7080 


7245 


7410 7575 


7741 


5 


9 


8071 


8236 


8401 


8567 


8732 


8897 


9062 9227 


9392 


5 


2630 


9723 


9888 


4200053 


0218 


4200383 


0548 


4200713 0878 


4201043 


5 


1 


4201374 


1539 


1704 


1869 


2034 


2199 


2364 2529 


2694 


5 


2 


3024 


3189 


3354 


3519 


3684 


3849 


4014 4179 


4344 


5 


3 


4674 


4838 


5003 


5168 


5333 


5498 


5663 5828 


5993 


5 


4 


6323 


6487 


6652 


6817 


6982 


7147 


7312 7477 


7641 


i 5 


5 


7971 


8136 


8301 


8465! 8630 


8795 


1 8960 9125 


9289 


5 


6 


9619 


9784 


9948 


/0113 
'l760 


4210278 


0442 


4210607 0772 


4210937 


5 


7 


4211266 


1431 


4211595 


1925 


2089 


i 2254 2419 


2583 


5 


8 


2913 


3077 


3242 


3406 


3571 


3736 


1 3900 4065 


4229 


5 


9 


4558 


4723 


4888 


5052 


5217 


5381 


5546 5710 


5875 


5 


^— 


1 


2 


3 


4: 


5 


G 


1 T 8 


9 


= 



46 


LOGARITHMS 


OF NUMBERS 


FROM 1 TO 36,000. [Tfl^/ei.!! 


Between 26400 = log. ~ 


1 4-4216039, and 27000 = log. -i 4-4313638. 1 


tens. 


1 


2 


3 


4 


5 


t> 


T 8 


9 


dif\ 


2640 


4216204 


6368 


4216533 


6697 


4216862 


7026 


4217191 7355 


4217520 


164 


1 


7848 


8013 


8177 


8342 


8506 


8671 


8835 8999 


9164 


4 


2 


9493 


9657 


9821 


9986 


4220150 


0314 


4220479 0643 


4220807 


4 


3 


4221136 


1300 


4221465 


1629 


1793 


1957 


2122 2286 


2450 


4 


4 


2779 


2943 


3107 


3271 


3436 


3600 


3764 3928 4093' 


4 


5 


4421 


4585 


4749 


4913 


5078 


5242 


5406 5570 


5734 


4 


6 


6063 


6227 


6391 


6555 


6719 


6883 


7047 7211 


7375 


4 


7 


7703 


7868 


8032 


8196 


8360 


8524 


8688 8852 


9016 


4 


8 


9344 


9508 


9672 


9836 


4230000 


0164 


4230328 0492 


4230656 


4 


9 


4230984 


1147 


4231311 


1475 


1639 


1803 


1967 2131 


2295 


4 


2650 


2623 


2786 


2950 


3114 


3278 


3442 


3606 3770 


3933 


4 


1 


. 4261 


4425 


4589 


4753 


4916 


5080 


5244 5408 


5571 


4 


2 


5899 


6063 


6226 


6390 


6554 


6718 


6881 7045 


7209 


4 


3 


7536 


7700 


7864 


8027 


8191 


8355 


8518 8682 


8846 


4 


4 


9173 


9336 


9500 


9664^ 9827 


9991 


4240154 0318 


4240432 


4 


5 


4240809 


0972 


4241136 


1300 


4241463 


1627 


1790 1954 


2117 


4 


6 


2444 


2608 


2771 


2935 


3098 


3262 


3425 3589 


3752 


163 


7 


4079 


4242 


4406 


4569 


4733 


4896 


5060 5223 


5386 


3 


S 


5713 


5877 


6040 


6203 


6367 


6530 


6693 6357 


7020 


3 


9 


7347 


7510 


7673 


7837 


8000 


8163 


8327 8490 


8653 


3 


2660 


8980 


9143 


9306 


9469 


9633 


9796 


9959 /0122 
4251591 ' 1754 


4250286 


3 


1 


4250612 


0775 


4250938 


1102 


4251265 


1428 


1917 


3 


2 


2244 


2407 


2570 


2733 


2896 


3059 


3222 3385 


3549 


3 


3 


3875 


4038 


4201 


4364 


4527 


4690 


4853 5016 


5179 


3 


4 


5505 


5668 


5831 


5994 


6157 


6320 


6483 6646 


6809 


3 


5 


7135 


7298 


7461 


7624 


7787 


7950 


8113 8276 


8439 


3 


6 


8764 


8927 


9090 


9253 


9416 


9579 


9742 9904 


4260067 


3 


7 


4260393 


0556 


4260719 


0881 


4261044 


1207 


4261370 1533 


1695 


3 


8 


2021 


2184 


2347 


2509 


2672 


2335 


2998 3160 


3323 


3 


9 


3648 


3811 


3974 


4137 


4299 


4462 


4625 4787 


4950 


3 


2670 


5275 


5438 


5601 


5763 


5926 


6088 


6251 6414 


6576 


3 


1 


6901 


7064 


7227 


7389 


7552 


7714 


7877 8039 


8202 


3 


2 


8527 


8690 


8852 


9015 


9177 


9340 


9502 9665 


9827 


3 


3 


4270152 


0315 


4270477 


0639 


4270802 


0964 


4271127 1289 


4271452 


162 


4 


1776 


1939 


2101 


2264^ 2426 


2588 


2751 2913 


3076 


2 


5 


3400 


3563 


3725 


3887 


4050 


4212 


4374 4536 


4699 


2 


6 


5023 


5186 


5348 


5510 


5672 


5835 


5997 6159 


6321 


2 


7 


6646 


6808 


6970 


7133 


7295 


7457 


7619 7781 


7944 


2 


8 


8268 


8430 


8592 


8754 


8917 


9079 


9241 9403 


9565 


2 


9 


9889 


.0051 
1672 


4280213 


0376 


4280533 


0700 


4280862 1024 


4281186 


2 


2680 


4281510 


1834 


1996 


2158 


2320 


2482 2644 


2806 


2 


1 


3130 


3292 


3454 


3616 


3778 


3940 


4102 4264 


4426 


2 


2 


4750 


4912 


5073 


5235 


5397 


5559 


5721 5883 


6045 


2 


3 


6369 


6530 


6692 


6854 


7016 


7178 


7340 7501 


7663 


2 


4 


7987 


8149 


8311 


8472 


8634 


8796 


8958 9119 


9281 


2 


5 


9605 


9766 


9928 /0090 


4290252 


0413 


4290575 0737 


4290898 


2 


6 


4291222 


1383 


4291545 ' 


1707 


1868 


2030 


2192 2353 


2515 


2 


7 


2838 


3000 


3162 


3323 


3485 


3646 


3808 3969 


4131 


2 


8 


4454 


4616 


4777 


4939 


5100 


5262 


5423 5585 


5747 


2 


9 


6070 


6231 


6393 


6554 


6715 


6877 


7038 7200 


7361 


161 


2690 


7684 


7846 


8007 


8169 


8330 


8491 


8653 8814 


8976 




1 


9298 


9460 


9621 


9782 


9944 /0105 


4300267 0428 


4300589 




2 


4300912 


1073 


4301235 


1396 


4301557 ' 


1718 


1880 2041 


2202 




3 


2525 


2686 


2847 


3009 


3170 


3331 


3492 3653 


3815 




4 


4137 


4298 


4460 


4621 


4782 


4943 


5104 5265 


5427 


1,- 


5 


5749 


5910 


6071 


6232 


6393 


6554 


6715 6877 


7038 




6 


7360 


7521 


7682 


7843 


8004 


8165 


8326 8487 


8648 




7 


8970 


9132 


9293 


9454 


9615 


9776 


9937 /0098 
4311546 M707 


4310258 




8 


4310580 


0741 


4310902 


1063 


4311224 


1385 


1868 


1 


9 


2190 


2351 


2512 


2672 


2833 


2994 


3155 3316 


3477 


1 




1 


2 


3 


4 


5 


6 


1 8 


^ ^! 



Table I.] 


LOGARITHBIS 


OF NUMBERS 


FROM 1 TO 36,000. 47 1 


Between 27000 = log " 


1 4-4313638, and 27600 = log 


-14-4409091. 1 


tens. 


1 


2 


3 


4 


5 


6 


t 


8 


9 


dif.\ 


2700 


4313798 


3959 


4314120 


4281 


4314442 


4603 


4314763 


4924 


4315085 


161 


1 


5407 


5567 


5728 


5889 


6050 


6210 


6371 


6532 


6693 


1 


2 


7014 


7175 


7336 


7496 


7657 


7818 


7978 


8139 


8300 


1 


3 


8621 


8782 


8942 


9103 


9264 


9424 


9585 


9746 


9906 


1 


4 


4320227 


0388 


4320549 


0709 


4320870 


1030 


4321191 


1352 


4321512 1 II 


5 


1833 


1994 


2154 


2315 


2475 


2636 


2796 


2957 


3117 


1 


6 


3438 


3599 


3759 


3920 


4080 


4241 


4401 


4562 


4722 


160 


7 


5043 


5203 


5364 


5524 


5685 


5845 


6005 


6166 


6326 





1 8 


6647 


6807 


6968 


7128 


7288 


7449 


7609 


7769 


7930 





1 9 


8250 


8411 


8571 


8731 


8892 


9052 


9212 


9372 


9533 





2710 


9853 


/0013 
1616 


4330174 


0334 


4330494 


0654 


4330815 


0975 


4331135 





1 1 


4341455 


1776 


1936 


2096 


2256 


2416 


2577 


2737 





2 


3057 


3217 


3377 


3537 


3697 


3858 


4018 


4178 


4338 





3 


4658 


4818 


4978 


5138 


5298 


5458 


5613 


5778 


5938 





4 


6258 


6418 


6573 


6738 


6898 


7058 


7218 


7378 


7538 





5 


7858 


8018 


8178 


8338 


8498 


8658 


8818 


8978 


9138 





6 


9458 


9617 


9777 


9937 


4340097 


0257 


4340417 


0577 


4340737 





7 


4341056 


1216 


4341376 


1536 


1696 


1855 


2015 


2175 


2335 





8 


2654 


2814 


2974 


3134 


3293 


3453 


3613 


3773 


3932 





9 


4252 


4412 


4571 


4731 


4891 


5050 


5210 


5370 


5529 





2720 


5849 


6008 


6168 


6328 


6487 


6647 


6807 


6966 


7126 





1 


7445 


7605 


7764 


7924 


8083 


8243 


8403 


8562 


8722 





2 


9041 


9200 


9360 


9519 


9679 


9838 


9998 


/0157 
^1752 


4350317 





3 


4350636 


0795 


4350955 


1114 


4351274 


1433 


4351593 


1912 





4 


2230 


2390 


2549 


2709 


2868 


3028 


3187 


3346 


3506 


159 


5 


3824 


3984 


4143 


4303 


4462 


4621 


4781 


4940 


5099 


9 


6 


5413 


5577 


5736 


5896 


6055 


6214 


6374 


6533 


6692 


9 


7 


7011 


7170 


7329 


7488 


7643 


7807 


7966 


8125 


8284 


9 


8 


8603 


8762 


8921 


9080 


9240 


9399 


9558 


9717 


9876 


9 


9 


4360194 


0354 


4360513 


0672 


4360831 


0990 


4361149 


1308 


4361467 


9 


:2730 


1786 


1945 


2104 


2263 


2422 


2581 


2740 


2899 


3058 


9 


1 


3376 


3535 


3694 


3853 


4012 


4171 


4330 


4489 


4648 


9 


2 


4966 


5125 


5284 


5443 


5602 


5761 


5920 


6078 


6237 


9 


3 


6555 


6714 


6873 


7032 


7191 


7350 


7509 


7667 


7826 


9 


4 


8144 


8303 


8462 


8620 


8779 


8938 


9097 


9256 


9415 


9 


5 


9732 


9891 


4370050 


0208 


4370367 


0526 


4370685 


0843 


4371002 


9 


6 


4371320 


1478 


1637 


1796 


1955 


2113 


2272 


2431 


2589 


9 


7 


2907 


3065 


3224 


3383 


3541 


3700 


3859 


4017 


4176 


9 


8 


4493 


4652 


4810 


4969 


5127 


5286 


5445 


5603 


5762 


9 


9 


6079 


6237 


6396 


6555 


6713 


6872 


7030 


7189 


7347 


9 


2740 


7664 


7823 


7981 


8140 


8293 


8457 


8615 


6773 


8932 


158 


1 


9249 


9407 


9566 


9724 


9883 


/0041 
1625 


4380199 


0358 


4380516 


8 


2 


4380833 


0991 


4381150 


1308 


4381466 


1783 


1941 


2100 


8 


3 


2416 


2575 


2733 


2891 


3050 


3208 


3366 


3525 


3683 


8 


4 


3999 


4158 


4316 


4474 


4632 


4791 


4949 


5107 


5265 


8 


5 


5582 


5740 


5898 


6056 


6214 


6373 


6531 


6689 


6847 


8 


6 


7163 


7322 


7480 


7638 


7796 


7954 


8112 


8270 


8428 


8 


7 


8745 


8903 


9061 


9219 


9377 


9535 


9693 


9851 


4390009 


8 


8 


4390325 


0483 


4390641 


0799 


4390957 


1115 


4391273 


1431 


1589 


8 


9 


1905 


2063 


2221 


2379 


2537 


2695 


2853 


3011 


3169 


8 


2750 


3485 


3643 


3801 


3959 


4116 


4274 


4432 


4590 


4748 


8 


1 


5064 


5222 


5379 


5537 


5695 


5853 


6011 


6169 


6326 


8 


2 


6642 


6800 


6953 


7115 


7273 


7431 


7589 


7747 


7904 


8 


3 


8220 


8378 


8535 


8693 


8851 


9009 


9166 


9324 


9482 


8 


4 


9797 


9955 


4400112 


0270 


4400428 


0585 


4400743 


0901 


4401058 


8 


5 


4401374 


1531 


1689 


1847 


2004 


2162 


2319 


2477 


2635 


8 


6 


2950 


3107 


3265 


3422 


3580 


3738 


3895 


4053 


4210 


8 


7 


4525 


4683 


4840 


4993 


5155 


5313 


5470 


5628 


5785 


157 


8 


6100 


6258 


6415 


6572 


6730 


6887 


7045 


7202 


7360 


7 


9 


7674 


7832 


7989 


8147 


8304 


8461 


8619 


8776 


8933 


7 




1 


2 


3 


4 


5 


6 


T 


8 


» 





48 


LOGARITHMS 


OF NUMBERS FROM 1 TO 36,000. [Table I. 1 


Between 27600 = logr 


■1 4-4409091, and 28200 = log.-^ 4-4502491. 


tens. 


1 


2 


3 


4r 


5 


6 1 

/0035 
1608 


T 


8 


9 


dif. 


2760 


4409248 


9406 


4409563 


9720 


4409878 


4410192 


0349 


4410507 


157 


1 


4410821 


0979 


4411136 


1293 


4411450 ' 


1765 


1922 


2080 


7 


2 


2394 


2551 


2708 


2866 


3023 


3180 


3337 


3494 


3652 


7 1 


3 


3966 


4123 


4280 


4438 


4595 


4752 


4909 


5066 


5223 


7 ! 


4 


5533 


5695 


5852 


6009 


6166 


6323 


6480 


6637 


6794 


7 


5 


7108 


7265 


7423 


7580 


7737 


7894 


8051 


8208 


8355 


7 


6 


8679 


8836 


8993 


9150 


9307 


9464 


9621 


9778 


9935 


7 


7 


4420249 


0405 


4420562 


0719 


4420876 


1033 


4421190 


1347 


4421504 


7 


8 


1818 


1975 


2132 


2288 


2445 


2602 


2759 


2916 


3073 


7 


9 


3386 


3543 


3700 


3857 


4014 


4171 


4327 


4484 


4641 


7 


2770 


4954 


5111 


5268 


5425 


5582 


5738 


5895 


6052 


6209 


7 


1 


6522 


6679 


6835 


6992 


7149 


7306 


7462 


7619 


7776 


7 


2 


8089 


8246 


8402 


8559 


8716 


8372 


9029 


9135 


9342 


7 


3 


9655 


9812 


9969 


/0125 
' 1691 


4430282 


0438 


4430595 


0751 


4430908 


7 


4 


4431221 


1378 


4431534 


1847 


2004 


2160 


2317 


2473' 7 II 


5 


2786 


2943 


3099 


3256 3412 


3569 


3725 


3882 


4038 


156 


6 


4351 


4507 


4664 


4820 4977 


5133 


5290 


5446 


5602 


6 


7 


5915 


6072 


6228 


6384 6541 


6697 


6853 


7010 


7166 


6 


8 


7479 


7635 


7791 


7948 8104 


8260 


8417 


8573 


8729 


6 


9 


9042 


9198 


9354 


9511 9667 


9823 


9979 


,0136 


4440292 


6 


2780 


4440604 


0760 


4440917 


1073 4441229 


1385 


4441541/ 1698 


1354 


6 


1 


2166 


2322 


2478 


2635 2791 


2947 


3103 


3259 


3415 6 


2 


3727 


3883 


4040 


4196 4352 


4508 


4664 


4820 


4976 6 


3 


5288 


5444 


5600 


5756 5912 


6068 


6224 


6380 


6536 6 


4 


6848 


7004 


7160 


7316 7472 


7628 


7784 


7940 


8096 6 


5 


8408 


8564 


8720 


8876 


9032 


9188 


9343 


9499 


9655 

4451214 

2772 


6 


6 


9967 


/0123 
1681 


4450279 


0435 


4450590 


0746 


4450902 


1058 


6 


7 


4451526 


1837 


1993 


2149 


2305 


2460 


2616 


6 


8 


3083 


3239 


3395 


3551 


3706 


3862 


4018 


4174 


4329 


6 


9 


4641 


4797 


4952 


5108 


5264 


5419 


5575 


5731 


5886 


6 


2790 


6198 


6353 


6509 


6665 


6820 


6976 


7132 


7287 


7443 


6 


1 


7754 


7910 


8065 


8221 


8376 


8532 


8687 


8343 


8999 


6 


2 


9310 


9465 


9621 


9776 


9932 


/0087 
1642 


4460243 


0398 


4460554 


6 


3 


4460865 


1020 


4461176 


1331 


4461487 


1798 


1953 


2109 


155 


4 


2419 


2575 


2730 


2886 


3041 


3197 


3352 


3507 


3663 


5 


5 


3974 


4129 


4284 


4440 


4595 


4750 


4906 


6061 


5216 


5 


6 


5527 


5682 


5838 


5993 


6148 


6304 


6459 


6614 


6769 


5 


7 


7080 


7235 


7390 


7546 


7701 


7856 


8011 


8167 


8322 


5 


6 


8632 


8788 


8943 


9098 


9253 


9408 


9563 


9719 


9874 


5 


9 


4470184 


0339 


4470494 


0650 


4470805 


0960 


4471115 


1270 


4471425 


5 


2800 


1735 


1891 


2046 


2201 


2356 


2511 


2666 


2821 


2976 


5 


1 


3286 


3441 


3596 


3751 


3906 


4061 


4216 


4371 


4526 


5 


2 


4836 


4991 


5146 


5301 


5456 


5611 


5766 


5921 


6076 


5 


3 


6386 


6541 


6696 


6851 


7006 


7161 


7315 


7470 


7625 


& 


4 


7935 


8090 


8245 


8400 


8554 


8709 


8864 


9019 


9174 


5 


5 9483 


9638 


9793 


9948 


4480103 


0258 


4480412 


0567 


4480722 


5 


6 4481031 


1186 


4481341 


1496 


1650 


1805 


1960 


2115 


2269 


5 


7 2579 


2734 


2883 


3043 


3198 


3352 


3507 


3662 


3816 


5 


8 4126 


4280 


4435 


4590 


4744 


4899 


5054 


5208 


5363 


5 


9 


5672 


5827 


5981 


6136 


6290 


6445 


6600 


6754 


6909 


5 


2810 


7218 


7372 


7527 


7681 


7836 


7990 


8145 


8299 


8454 


5 


1 


8763 


8917 


9072 


9226 


9381 


9535 


9690 


9844 


9999 


154 


2 


4490308 


0462 


4490616 


0771 


4490925 


1080 


4491234 


1389 


4491543 




3 


1852 


2006 


2160 


2315 


2469 


2624 


2778 


2932 


3087 




4 


1 3395 


3550 


3704 


3858 


4013 


4167 


4321 


4475 


4630 




5 4938 


5093 


5247 


5401 


5555 


5710 5864 


6018 


6172 




6 6481 


6635 


6789 


6943 


7098 


7252 7406 


7560 


7714 




7 -8023 


8177 


8331 


8485 


8639 


8793 8948 


9102 


9256 




8 9564 


9718 


9872 


/ C026 
1567 


4500180 


0334 4500439 


0643 


4500797 




9 4501105 


1259 


4501413 
3 


1721 


1875 2029 


2183 


2337 




1 


2 


4 1 5 


6 T 


■ II Hi 


?> 


___ 



Table i.] logarithms 


OF NUMBERS 


FROM 1 TO 36,000. 49 


Between 28200== log. - 


1 4-4502491, and 28800 = log "i 4-4593925. || 


tens. 


1 3 


3 


4 


5 


6 


T 8 


9 


dif. 


2820 


4502645 2799 


4502953 


3107 


4503261 


3415 


4503569 3723 


4503877 


154 


1 


4185 4339 


4493 


4647 


4801 


4954 


5108 5262 


5416 


4i 


2 


5724 5878 


6032 


6186 


6340 


6493 


6647 6801 


6955 


4 


3 


7263 7416 


7570 


7724 


7878 


8032 


8186 8339 


8493 


4 


4 


8801 8954 


9108 


9262 


9416 


9570 


9723 9877 


4510031 


4 


5 


4510338 0492 


4510646 


0799 


4510953 


1107 


4511261 1414 


1568 


4 


6 


1875 2029 


2183 


2336 


2490 


2644 


2797 2951 


3104 


4 


7 


3412 3565 


3719 


3873 


4026 


4180 


4333 4487 


4640 


4 


8 


4948 5101 


5255 


5408 


5562 


5715 


6869 6022 


6176 


153 


9 


6483 6636 


6790 


6943 


7097 


7250 


7404 7557 


7711 


3 


2830 


8018 8171 


8325 


8478 


8632 


8785 


8938 9092 


9245 


3 


1 


9552 9705 


9859 


/0012 
M546 


4520166 


0319 


4520472 0626 


4520779 


3 


2 


4521086 1239 


4521393 


1699 


1853 


2006 2159 


2312 


3 


3 


2619 2772 


2926 


3079 


3232 


3385 


3539 3692 


3845 


3 


4 


4152 4305 


4458 


4611 


4765 


4918 


5071 5224 


5377 


3 


5 


5684 5837 


5990 


6143 


6297 


6450 


6603 6756 


6909 


3 


6 


7215 7369 


7522 


7675 


7828 


7981 


8134 8287 


8440 


3 


7 


8746 8900 


9053 


9206 


9359 


9512 


9665 9818 


. 9971 


3 


8 


4530277 0430 


4530583 


0736 


4530889 


1042 


4531195 1348 


4531501 


3 


9 


1807 1960 


2113 


2266 


2419 


2572 


2725 2878 


3030 


3 


2840 


3336 3489 


3642 


3795 


3948 


4101 


4254 4407 


4559 


3 


1 


4865 5018 


5171 


5324 


5477 


5629 


5782 5935 


6088 


3 


2 


6394 6546 


6699 


6852 


7005 


7158 


7310 7463 


7616 


3 


3 


7921 8074 


8227 


8380 


8532 


8685 


8838 8990 


9143 


3 


4 


9449 9601 


9754 


9907 


4540059 


0212 


4540365 0517 


4540670 


3 


5 


4540975 1128 


4541281 


1433 


1586 


1739 


1891 2044 


2196 


3 


6 


2502 2654 


2807 


2959 


3112 


3264 


3417 3570 


3722 


3 


7 


4027 4180 


4332 


4485 


4637 


4790 


4942 5095 


5247 


3 


8 


5552 5705 


5857 


6010 


6162 


6315 


6467 6620 


6772 


152 


9 


7077 7229 


7382 


7534 


7687 


7839 


7991 8144 


8296 


2 


2850 


8601 8753 


8906 


9058 


9210 


9363 


9515 9668 


9820 


2 


1 


4550125 0277 


4550429 


0581 


4550734 


0886 


4551038 1191 


4551343 


2 


2 


1647 1800 


1952 


2104 


2257 


2409 


2561 2713 


2865 


2 


3 


3170 3322 


3474 


3627 


3779 


3931 


4083 4235 


4388 


2 


4 


4692 4844 


4996 


5148 


5300 


5453 


5605 5757 


5909 


2 


5 


6213 6365 


6517 


6670 


6822 


6974 


7126 7278 


7430 


2 


6 


7734 7886 


8038 


8190 


8342 


8494 


8646 8798 


8950 


2 


7 


9254 9406 


9558 


9710 


9862 


.0014 


4560166 0318 


4560470 


2 


■ 8 


4560774 0926 


4561078 


1230 


4561382 / 1534 


1686 1838 


1990 


2 


9 


2293 2445 


2597 


2749 


2901 


3053 


3205 3357 


3508 


2 


2860 


3812 3964 


4116 


4268 


4420 


4571 


4723 4875 


5027 


2 


1 


5330 5482 


5634 


5786 


5938 


6089 


6241 6393 


6545 


2 


2 


6848 7000 


7152 


7303 


7455 


7607 


7758 7910 


8062 


2 


3 


8365 8517 


8669 


8820 


8972 


9124 


9275 9427 


9578 


2 


4 


9882/0033 
4571398^ 1549 


4570185 


0337 


4570488 


0640 


4570791 0943 


4571095 


2 


5 


1701 


1853 


2004 


2156 


2307 2459 


2610 


2 


6 


2913 3065 


3216 


3368 


3519 


3671 


3822 3974 


4125 


2 


7 


4428 4580 


4731 


4883 


5034 


5186 


5337 5489 


5640 


151 


8 


5943 6094 


6246 


6397 


6549 


6700 


6851 7003 


7154 




9 


7457 7608 


7760 


7911 


8062 


8214 


8365 8516 


8668 




2870 


8970 9122 


9273 


9424 


9576 


9727 


9878,0029 
4581391 / 1542 


4580181 


1 


1 


4580483 0634 


4580786 


0937 


4581088 


1239 


1693 




2 


1996 2147 


2298 


2449 


2600 


2752 


2903 3054 


3205 


1 


3 


3507 3659 


3810 


3961 


4112 


4263 


4414 4565 


4717 




4 


5019 5170 


5321 


5472 


5623 


5774 


5925 6076 


6227 


1 


5 


6530 6681 


6832 


8983 


7134 


7285 


7436 75871 


7738 




6 


8040 8191 


8342 


8493 


8644 


8795 


8946 9097 


9248 




7 


9550 9701 


9851/ 
4591361 ' 


0002 


4590153 


0304 


4590455 0606 


4590757 




8 


4591059 1210 


1511 


1662 


1813 


1964 2115 


2266 




9 


2567 2718 


2869 


3020 


3171 


3322 


3472 3623 


3774 






1 2 


3 


4 


5 


6 


T 8 


^ 






5 






G 











50 


LOGARITHMS < 


3F NUMBERS FROM 1 TO 36,000. [Table I. 1 


Between 28800 = log.~ 


1 4-4593925, and 29400 = log 


.-1 4-4683473. j 


tens. 


1 


2 s 


4 5 

4528 4594679 


^ i 


T 


8 


9 


dif- 


2880 


4594076 


4226 


4594377 


4830 


4594980 


5131 


4595282 


151 


1 


5582 


5734 


5885 


6036 


6186 


6337 


6488 


6638 


6789 




2 


7090 


7241 


7392 


7542 


7693 


7844 


7994 


8145 


8296 




3 


8597 


8748 


8898 


9049 


9200 


9350 


9501 


9651 


9802 




4 


4600103 


0254 


4600404 


0555 


4600705 


0856 


4601007 


1157 


4601308 




5 


1609 


1759 


1910 


2060 


2211 


2361 


2512 


2662 


2813 


150 


6 


3114 


3264 


3415 


3565 


3716 


3866 


4017 


4167 


4317 





7 


4618 


4769 


4919 


5070 


5220 


5370 


5521 


5671 


5822 





8 


6122 


6273 


6423 


6573 


6724 


6874 


7024 


7175 


7325 





9 


7626 


7776 


7926 


8077 


8227 


8377 


8528 


8678 


8828 





2890 


9129 


9279 


9429 


9579 


9730 


9880 


4610030 


0180 


4610331 





1 


4610631 


0781 


4610932 


1082 


4611232 


1382 


1532 


1683 


1833 





2 


2133 


2283 


2433 


2584 


2734 


2884 


3034 


3184 


3334 





3 


3634 


3785 


3935 


4085 


4235 


4385 


4535 


4685 


4835 





4 


5135 


5285 


5435 


5585 


5736 


5886 


6036 


6186 


6336 





5 6636 


6786 


6936 


7086 


7236 


7386 


7536 


7686 


7836 


-0 


6 8136 


8285 


8435 


8585 


8735 


8885 


9035 


9185 


9335 





7 9635 


9785 


9935/ 
4621433^ 


0085 


4620234 


0384 


4620534 


0684 


4620834 





8 4621134 


1284 


1583 


1733 


1883 


2033 


2183 


2332 





9 


2632 


2782 


2932 


3081 


3231 


3381 


3531 


3680 


3830 





2900 


4130 


4279 


4429 


4579 


4729 


4878 


5028 


5178 


5328 





1 


5627 


5777 


5926 


6076 


6226 


6375 


6525 


6675 


6824 





2 


7124 


7273 


7423 


7573 


7722 


7872 


8022 


8171 


8321 





3 


8620 


8770 


8919 


9069 


9218 


9368 


9517 


9667 


9817 





4 4830116 


0265 4630415 


0564 4630714 


086334631013 


1162 


4631312^149 !S 


5 


1611 


1760 


1910 


2059 


2209 


2358 


2508 


2657 


2807 


9 


6 


3106 


3255 


3404 


3554 


3703 


3853 


4002 


4152 


4301 


9 


7 


4600 


4749 


4898 


5048 


5197 


5347 


5496 


5645 


5795 


9 


8 


6093 


6243 


6392 


6541 


6691 


6840 


6989 


7139 


7288 


9 


9 


7587 


7736 


7885 


8034 


8184 


8333 


8482 


8631 


8781 


9 


2910 


9079 


9228 


9378 


9527 


9676 


9825 


9974/0124 


4640273 


9 


1 


4640571 


0720 


4640870 


1019 


4641168 


1317 


4641466 ' 


1615 


1765 


9 


2 


2063 


2212 


2361 


2510 


2659 


2808 


2958 


3107 


3256 


9 


3 


3554 


3703 


3852 


4001 


4150 


4299 


4448 


4597 


4746 


9 


4 


5045 


5194 5343 


5492^ 5641 


5790 


5939 


6088 


6237 


9 


5 


6535 


6684 


6833 


6981 


7130 


7279 


7428 


7577 


7726 


9 


6 


8024 


8173 


8322 


8471 


8620 


8769 


8918 


9067 


9215 


9 


7 


9513 


9662 


9811 


9960 


4650109 


0258 


4650406 


0555 


4650704 


9 


8 


4651002 


1151 


4651299 


1448 


1597 


1746 


1895 


2043 


2192 


9 


9 


2490 


2639 


2787 


2936 


3085 


3234 


3382 


3531 


3680 


9 


2920 


3977 


4126 


4275 


4423 


4572 


4721 


4870 


5018 


5167 


9 


1 


5464 


5613 


5762 


5910 


6059 


6208 


6356 


6505 


6653 


9 


2 


6951 


7099 


■ 7248 


7397 


7545 


7694 


7842 


7991 


8140 


9 


3 


8437 


8585 


8734 


8882 


9031 


9180 


9328 


9477 


9625 


9 


4 


9922 / 0071 


4660219 


0368 


4660516 


0665 


4660813 


0962 


4661110 


9 


5 


4661407^ 


1556 


1704 


1853 


2001 


2149 


2298 


2446 


2595 


148 


6 


2892 


3040 


3188 


3337 


3485 


3634 


3782 


3930 


4079 


8 


7 


4376 


4524 


4672 


4821 


4969 


5117 


5266 


5414 


5562 


8 


8 


5859 


6007 


6156 


6304 


6452 


6601 


6749 


6897 


7045 


8 


9 


7342 


7490 


7639 


7787 


7935 


8083 


8232 


8380 


8528 


8 


2930 


8824 


8973 


9121 


9269 


9417 


9565 


9714 


9862 


4670010 


8 


I 


4670306 


0455 


4670603 


0751 


4670899 


1047 


4671195 


1343 


1492 


8 


2 


1788 


1936 


2084 


2232 


2380 


2528 


2676 


2824 


2973 


8 


3 


3269 


3417 


3565 


3713 


3861 


4009 


4157 


4305 


4453 


8 


4 


4749 


4897 


5045 


5193 


5341 


5489 


5637 


5785 


5933 


8 


5 


6229 


6377 


6525 


6673 


6821 


6969 


7117 


7265 


7413 


8 


6 


7708 


7856 


8004 


8152 


8300 


8448 


8596 


8744 


8892 


8 


7 


9187 


9335 


9483 


9631 


9779 


9927 


4680074 


0222 


4680370 


8 


8 


4680666 


0814 


4680961 


1109 


4681257 


14Q5 


1553 


1700 


1848, 


8 


9 


2144 


2291 


2439 


2587 


2735 


2882 


3030 


3178 


3326 


8 


I 1 


2 1 3 


4 


5 


6 


T 


8 


9 





: Tahk I.] 


LOGARITHMS OF NUMBERS 


FROM J TO 36,000. 51 


Betw-een 29400 ^ log"' 4-4r)83473, an 


d 30000 = lo,2- .-' 4-4771213. 


tens. 


1 


« 9 3 4 i 5 

3769 '4683916 406414684212 


il ? S f 9 


dif. 


2940 


4683621 


4360 4684507 4655^694803 


148 


1 


5098 


5246 


5393 5541 5689 


5836 5984 613U G279 
7312 7460 7607 7755 
8788 8935 90831 9231 


8 


2 


6574 


6722 


6870 7017 7165 


8 


3 


8050 


8198 


8345 8493 8840 


8, 


4 952S 


9673 


9821 9368M690116 


0263 4690411 0558 '4690706 


147 


5 


4691000 


1148 


4691295 1443 


1590 


1738 


1885 2033 2180 


7 


6 


2475 


2622 


2770 2917 


3064 


3212 


3359 3507 


3654 


7 


7 


3949 


4096 


4243 4391 


4538 


4685 


4833 4980 


5127 


7 


8 


5422 


5569 


5717 5864 


6011 


6159 


6306 6453 


6600 


7 


9 


6895 


7042 


7190 7337 


7484 


7631 


7778 7926 


8073 


7 


2950 


8367 


8515 


8662 8809 


8956 


9103 


9251 9398 


9545 


7 


1 


9839 


9986 


4700134 0281 


4700428 


0575 


4700722 0869 


4701016 


7 


2 


4701311 


1458 


1605 1752 


1899 


2046 


2193 2340 


2487 


7 


3 


2782 


2929 


3076 3223 


3370 


3517 


3664 3811 


3958 


7 


4^ 4252 


4399 


4546 4693! 4840 


4987^ 5134 5281' 5428 


7 


5 


5722 


5869 


6016 6163 


6310 


6457 


6604 6750 


6897 


7 


6 


7191 


7338 


7485 7632 


7779 


7926 


8073 8219 


8366 


7 


7 


8660 


8807 


8954 9101 


9248 


9394 


9541 9688 


9835 


7 


8 


4710129 


0275 


4710422 0569 


4710716 


0863 


4711009 il56 


4711303 


7 


9 


1596 


1743 


1890 2037 


2183 


2330 


2477 2624 


2770 


7 


2960 


3064 


3211 


3357 3504 


3651 


3797 


3944 4091 


4237 


7 


1 


4531 


4677 


4824 4971 


5117 


5264 


5411 5557 


5704 


7 


2 


5997 


6144 


6290 6437 


6584 


6730 


6877 7023 


7170 


7 


1 3 


7463 


7610 


7756 7903 


8049 


8196 


8342 8489 


8635 


7 


! 4' 8929 


9075 


9222 9368 


9515 


9661 


9808 9954M720101 


146 


5 


4720393 


0540 


4720686 0833 


4720979 


1126 


4721272 1419 


1565 


6 


6 


1858 


2004 


2151 2297 


2444 


2590 


■ 2736 2883 


3029 


6 


7 


3322 


3468 


3615 3761 


3907 


4054 


4200 4346 


4493 


6 


8 


4785 


4932 


5078 5224 


5371 


5517 


5663 5809 


5956 


6 


9 


6248 


6395 


6541 6687 


6833 


6980 


7126 7272 


7418 


6 


2970 


7711 


7857 


8003 8149 


8296 


8442 


8588 8734 


8880 


6 


1 


9173 


9319 


9465 9611 


9757 


9903 


4730050 0196 


4730342 


6 


2 


4730634 


0780 


4730926 1073 


4731219 


1365 


1511 1657 


1803 


6 


3 


2095 


2241 


2387 2533 


2679 


2825 


2972 3118 


3264 


6 


4 


3556 


3702 


3848 3994 


4140 


4286 


4432 4578 


4724 


6 


5 


5016 


5162 


5308 5454 


5600 


5746 


5891 6037 


6183 


6 


6 


6475 


6621 


6767 6913 


7059 


7205 


7351 7497 


7642 


6 


7 


7934 


8080 


8226 8372 


8518 


8664 


8809 8955 


9101 


6 


8 


9393 


9539 


9684 9830 


9976 


/0122 

1580 


4740268 0413 


4740559 


6 


9 


4740851 


0997 


4741142 1288 


4741434 ^ 


1725 1871 


2017 


6 


2980 


2308 


2454 


2600 2746 


2891 


3037 


3183 3328 


3474 


6 


1 


3765 


3911 


4057 4202 


4348 


4494 


4639 4785 


4931 


6 


2 


5222 


5368 


5513 5659 


5805 


5950 


6096 6241 


6387 


6 


3 


6678 


6824 


6969 7115 


7260 


7406 


7552 7697 


7843 


6 


4 


8134 


8279 


8425 8570 


8716 


8861 


9007 9152 


9298 


145 


5 9589 


9734 


9880 / 0025 
4751334 ' 1480 


4750171 


0316 


4750462 0607 


4750753 5 |: 


6 


4751043 


1189 


1625 


1771 


1916 2061 


2207 


5 


7 


2498 


2643 


2788 2934 


3079 


3225 


3370 3515 


3661 


5 


8 


3951 


4097 


4242 4387 


4533 


4678 


4823 4969 


5114 


5 


9 


,5404 


5550 


5695 5840 


5986 


6131 


6276 6421 


6567 


5 


2990 


6857 


7002 


7148 7293 


7438 


7583 


7729 7874 


8019 


5 


1 


8309 


8455 


8600 8745 


8890 


9035 


9180 9326 


9471 


5 


2 


9761 


9906 


4760051 0196 


4760342 


0487 


4760632 0777 


4760922 


5 


3 


4761212 


1357 


1502 1648 


1793 


1938 


2083 2228 


2373 


5 


4 


2663 


2808 


2953 3098 


3243 


3388 


3533 3678 


3823 


5 


5 


4113 


4258 


4403 4548 


4693 


4838 


4983 5128 5273 


5 


6 


5563 


5708 


5853 5998 


6143 


6288 


6433 6578 6723 


5 


7 


7012 


7157 


7302 7447 


7592 


7737 


7882 8027 8171 


5 


8 


8461 


8606 


8751 8896 


9041 


9185 


9330 9475 9620 


5 


9 


9909; 


r0054 


4770199 0344 


4770189 


0633 


4770778 0923 4771068 


5 




2 


3 4 


5 


6 


T .8 JI 



52 


LOGARITHMS 


OF NUMBERS 


FROM 1 TO 36,000. [Table I.I 




Between 30000 = log." 


■1 4-4771213, and 30600 = log. "^ 4-4857214. i 




tens. 


1 


2 


3 


4 


5 


6 


T 8 


9 


dif.\ 




3000 


4771357 


1502 


4771647 


1792 


4771936 


2031 


4772226 2371 


4772515 


145 




1 


2305 


2949 


3094 


3239 


3333 


3528 


3673 3313 


3962 


5 




2 


4252 


4396 


4541 


4686 


4830 


4975 


5119 5264 


5409 


5 




3 


5693 


5843 


5987 


6132 


6276 


6421 


6566 6710 


6355 


5 




4 


7144 


7288 


7433 


7578 


7722 


7367 


8011 8156 


8300 


5 




5 


8539 


8734 


8878 


9023 


9167 


9312 


9456 9601 


9745 


144 




6 


4730034 


0179 


4780323 


0463 


4780612 


0757 


4730901 1045 


4731190 


4 




7 


1479 


1623 


1763 


1912 


2056 


2201 


2345 2490 


2634 


4 




8 


2923 


3067 


3211 


3356 


3500 


3645 


3739 3933 


4073 


4 




9 


4366 


4511 


4655 


4799 


4943 


5033 


5232 5376 


5521 


4 




3010 


5309 


5954 


6093 


6242 


6386 


6531 


6675 6319 


6963 


4 




1 


7252 


7396 


7540 


7684 


7829 


7973 


8117 8261 


6405 


4 




2 


8694 


8838 


8932 


9126 


9271 


9415 


9559 9703 


9847 


4 




3 


4790135 


0230 


4790424 


0563 


4790712 


0355 


4791000 1144 


4791233 


4 




4 


1577 


1721 


1365 


2009 


2153 


2297 


2441 2565 


2729 


4 




5 


3017 


3161 


3305 


3449 


3593 


3737 


3331 4025 


4169 


4 




6 


4457 


4601 


4745 


4839 


5033 


5177 


5321 5465 


5609 


4 




7 


5897 


6041 


6185 


6329 


6473 


6617 


6761 6905 


7043 


4 




8 


7336 


7480 


7624 


7763 


7912 


8056 


8200 8343 


8487 


4 




9 


8775 


8919 


9063 


9207 


9350 


9494 


9633 9782 


9926 


4 




3020 


4800213 


0357 


4800501 


0645 


4800733 


0932 


4801076 1220 


4801363 


4 




1 1 


1651 


1795 


1939 


2082 


2226 


2370 


2513 2657 


2301 


4 




2 


3083 


3232 


3376 


3519 


3663 


3807 


3950 4094 


4233 


4 




3 


4525 


4669 


4812 


4956 


5100 


5243 


5337 5531 


5674 


4 




4 


5961 


6105 


6249 


6392 


6536 


6679 


6323 6967 


7110 


4 




5 


7397 


7541 


7684 


7828 


7972 


8115 


8259 8402 


8546 


4 




6 


8333 


8976 


9120 


9263 


9407 


9550 


9694 9337 


9931 


143 




7 


4810268 


0411 


4810555 


0698 


4810842 


0935 


4311128 1272 


4811415 


3 




8 


1702 


1846 


1989 


2132 


2276 


2419 


2563 2706 


2849 


3 




9 


3136 


3279 


3423 


3566 


3710 


3353 


3996 4140 


4233 


3 




3030 


4570 


4713 


4856 


5000 


5143 


5236 


5429 5573 


5716 


3 




1 


6003 


6146 


6289 


6432 


6576 


6719 


6362 7005 


7149 


3 




2 


7435 


7578 


7722 


7865 


6003 


8151 


8295 8438 


8531 


3 




3 


8367 


9010 


9154 


9297 


9440 


9583 


9726 9869 


4820013 


3 




4 


4820299 


0442 


4820535 


0728 


4820371 


1015 


4821153 1301 


1444 


3 




5 


1730 


1873 


2016 


2159 


2302 


2445 


2539 2732 


2375 


3 




6 


3161 


3304 


3447 


2590 


3733 


3876 


4019 4162 


4305 


3 




7 


4591 


4734 


4877 


5020 


5163 


5306 


5449 5592 


5735 


3 




8 


6021 


6164 


6307 


6449 


6592 


6735 


6878 7021 


7164 


3 




9 


7450 


7593 


7736 


7879 


8021 


8164 


8307 8450 


8593 


3 




3040 


6379 


9022 


9164 


9307 


9450 


9593 


9736 9379 


4830021 


3 




1 


4830307 


0450 


4830593 


0735 


4830378 


1021 


4831164 1307 


1449 


3 




2 


1735 


1873 


2020 


2163 


2306 


2449 


2591 2734 


2877 


3 




3 


3162 


3305 


3448 


3590 


3733 


3876 


4018 4161 


4304 


3 




4 


4539 


4732 


4874 


5017 


'5160 


5302 


5445 5533 


5730 


3 




5 


6016 


6158 


6301 


6443 


6536 


6729 


6371 7014 


7156 


3 




6 


7442 


7584 


7727 


7869 


8012 


8154 


8297 6439 


8532 


3 




7 


8367 


9010 


9152 


9295 


9437 


9580 


9722 9865 


4840007 


3 




8 


4340292 


0435 


4340577 


0720 


4840862 


1004 


4841147 1289 


1432 


142 




9 


1717 


1359 


2002 


2144 


2286 


2429 


2571 2714 


2856 


2 




3050 


3141 


3283 


3426 


3568 


3710 


3353 


■•3995 4137 


4230 


2 




1 


4564 


4707 


4849 


4991 


5134 


5276 


5418 5561 


5703 


2 




2 


5933 


6130 


6272 


6414 


6557 


6699 


6841 6934 


7126 


2 




3 


7410 


7553 


7695 


7337 


7979 


8121 


8264 8406 


8548 


2 




4 


8333 


8975 


9117 


9259 


9401 


9543 


9686 9828 


9970 


2 




5 


4850254 


0396 


4350539 


0681 


4850323 


0965 14851107 1249 


48513911 2 




6 


1676 


1813 


1960 


2102 


2244 


2336 


2528 2670 


2812 


2 




7 


3096 


3239 


3331 


3523 


3665 


3307 


3949 4091 


4233 


2 




S 


4517 


4659 


4301 


4943 


5035 


5227 


5369 5511 


5653 


2 




9 

±==55 


5937 


6079 


6221 


6363 


6505 


6647 


6788 6930 


7072 






1 


2 


3 


4 


S 


6 


T 8 


9 







1 Table I.] 


LOGARITHMS OF NUMBERS 


FROM 1 TO 36,000. 53 


Between 30600 = log."' 4-4857214, and 31200 = log."! 4-4941546. | 


te-ns. 


1 


2 3 4: 


5 


6 


T 


8 


9 


dif. 


13060 


4357356 


7498 4857640 7782 


4857924 


8066 


4858208 


8350 


4858491 


142 


1 


8775 


8917 9059 9201 


9343 


9484 


9626 


9768 


9910 


2 


2 


4860194 


0336 4860477 0619 
1754 1895 2037 
3171 3313 3455 


4860761 


0903 


4861045 


1186 


4861328 


2 


3 


1612 


2179 


2321 


2462 


2604 


2746 


2 


4 


3029 


3596 


3738 


3880 


4021 


4163 


2 


5 


4446 


4583 


4730 4872 


5013 


5155 


5297 


5438 


5580 


2 


6 


5863 


6005 


6146 6288 


6430 


6571 


6713 


6855 


6996 


2 


7 


7279 


7421 


7563 7704 


7846 


7987 


8129 


8270 


8412 


2 


8 


8695 


8837 


8978 9120 


9261 


9403 


9544 


9686 


9827 


2 


9 


4870110 


0252 


4870393 0535 


4870676 


0818 


4870959 


1101 


4871242 


2 


3070 


1525 


1667 


1808 1950 


2091 


2232 


2374 


2515 


2657 


2 


1 


2940 


3081 


3222 3364 


3505 


3647 


3788 


3929 


4071 


2 


2 


4353 


4495 


4636 4778 


4919 


5060 


5202 


5343 


5484 


2 


3 


5767 


5908 


6050 6191 


6332 


6473 


6615 


6756 


6897 


2 


4 


7180 


7321 


7462 7604 


7745 


7886 


8027 


8169 


8310 


2 


5 


8592 


8734 


8875 9016 


9157 


9299 


9440 


9581 


9722 


2 


6 


4830004 


0146 


4880287 0428 


4880569 


0710 


4880852 


0993 


4881134 


2 


7 


1416 


1557 


1698 1839 


1981 


2122 


2263 


2404 


2545 


2 


8 


2827 


2968 


3109 3251 


3392 


3533 


3674 


3815 


3956 


2 


9 


4238 


4379 


4520 4661 


4802 


4943 


5084 


5225 


5366 


141 


3080 


5648 


5789 


5930 6071 


6212 


6353 


6494 


6635 


6776 




1 


7058 


7199 


7340 7481 


7622 


7763 


7904 


8045 


8185 




2 


8467 


8608 


8749 8890 


9031 


9172 


9313 


9454 


9594 




3 


9876 


/0017 


4890158 0299 


4890440 


0580 


4890721 


0862 


4861003 




4 


4891285/' 1425" 1566 1707 


1848 


1989 


2129 


2270 


2411 




5 


2692 


2833 


2974 3115 


3256 


3396 


3537 


3678 


3818 




6 


4100 


4241 


' 4381 4522 


4663 


4804 


4944 


5085 


5226 




7 


5507 


5648 


5788 5929 


6070" 


6210 


6351 


6492 


6632 




8 


6914 


7054 


7195 7335 


7476 


7617 


7757 


7898 


8038 




9 


8320 


8460 


8601 8741 


8882 


9023 


9163 


9304 


9444 




3090 


9725 


9866 


4900006 0147 


4900287 


0428 


4900569 


0709 


4900850 




1 


4901131 


1271 


1412 1552 


1693 


1833 


1973 


2114 


2254 




2 


2535 


2676 


2816 2957 


3097 


3238 


3378 


3518 


3659 




3 


3940 


4080 


4220 4301 


4501 


4642 


4782 


4922 


5063 




4 


5343 


5484 


5624 5765 


5905 


6045 


6186 


6326 


6466 


140 


5 


6747 


6887 


7027 7168 


7308 


7448 


7589 


7729 


7869 





6 


8150 


8290 


8430 8571 


8711 


8851 


8991 


9132 


9272 





7 


9552 


9693 


9833 9973 


4910113 


0253 


4910394 


0534 


4910674 





8 


4910954 


1094 


4911235 1375 


1515 


1655 


1795 


1935 


2076 





9 


2356 


2496 


2636 2776 


2916 


3057 


3197 


3337 


3477 





3100 


3757 


3897 


4037 4177 


4317 


4457 


4597 


4738 


4878 





1 


5158 


5298 


5438 5578 


5718 


5858 


5998 


6138 


6278 





2 


6558 


6698 


6838 6978 


7118 


7258 


7398 


7538 


7678 





3 


7958 


8098 


8238 8378 


8517 


8657 


8797 


8937 


9077 





4 


9357 


9497 


9637 9777 


9917 


/0057 


4920196 


0336 


4920476 





5 


4920756 


0896 


4921036 1175 


4921315 > 14551 


1595 


1735 


1875 





6 


2154 


2294 


2434 2574 


2714 


-2853 


2993 


3133 


3273 


0. 


7 


3552 


3692 


3832 3972 


4111 


4251 


4391 


4531 


4670 





8 


4950 


5090 


5229 5369 


5509 


5648 


5788 


5928 


6068 





9 


6347 


6487 


6626 6766 


6906 


7045 


7185 


7325 


7464 





3110 


7744 


7883 


8023 8162 


8302 


8442 


8581 


8721 


8861 





1 


9140 


9279 


9419 9558 


9698 


9838 


9977 
4931373 1 


r0117 


4930256 





2 


4930535 


0675 


4930815 0954 


4931094 


1233 


1512 


1652 





3 


1931 


2070 


2210 2349 


2489 


2628 


2768 


2907 


3047 





4 


3326 


3465 


3604 3744 


3883 


4023 


4162 


4302 


4441 





5 


4720 


4859 


4999 5138 


5278 


5417 


5556 


5696 


5835 


' 


6 


6114 


6253 


6393 6532 


6671 


6811 


6950 


7089 


7229 





7 


7507 


7647 


7786 7925 


8065 


8204 


8343 


8483 


8622 





8 


8900 


9040 


9179 9318 


9457 


9597 


9736 


9875 


4940015 





9 


4940293 


0432 


4940571 0711 


4940850 


0989 


4941123 


1268 


1407 







1 


2 


3 4 


5 


6 


T 


8 


9 


1 



5* 



|54 


LOGARITHMS OF NUMBERS FROM 1 TO 36,000. [Table I. Ifl 


Between 31200 = log." 


~' 4-4941546, and 31800 = log."' 4-5024271. || 


tens. 


1 


2 I 3 


4 F 5 


6 i 7 S f 9 


dif. 


3120 


4941685 


1824 4941964 


2103 4942242 


2381 4942520 2659^4942799 


140 


1 


3077 


3216 3355 


3494 3633 


3773! 3912 4051 j 4190 


139 


2 


4468 


4607 4746 


4885 5024 


5164 


5303 5442 i 5581 


9 


3 


5859 


5998 


6137 


6276 6415 


6554 


6693 68321 6971 


9 


4 


7249 


7388 


i 7527 


7666* 7805 


7944 


8083 8222^ 8361 


9 


5 


8639 


8778 


8917 


9056 


■ 9195 


9334 


9473 9612J 9751 


9 


6 


4950029 


0168 


4950307 


0445 


4950584 


0723 


4950862 100154951140 


9 • 


7 


1418 


1557 


1695 


1834 


1973 


2112 


2251 2390 1 2529 


9 


8 


2806 


2945 


3084 


3223 


3362 


3500 


3639 37781 3917 


9 


9 


4194 


4333 


4472 


4611 


4750 


4838 


5027 5166 5305 


9 


3130 


5582 


5721 


5860 


5998 


6137 


6276 


6415 6553 6692 


9 


1 


6969 


7108 


7247 


7385 


7524 


7663 


7802 7940 8079 


9 


2 


8356 


8495 


8634 


8772 


8911 


9049 


9138 93271 9465 


9 


3 


9743 


9881 


4960020 


0158«j4960297 


0436 


4950574 071314960851 


9 


4 


4961128 


1267 


1406 


15448 1583 


1821' I960 2098^ 2237 


9 


5 


2514 


2653 


2791 


2930 


3063 


3207 


3345 3484 3622 


9 


6 


3899 


4038 


4176 


4314 


4453 


4591 


4730 4868 5007 


9 


7 


5284 


5422 


5560 


5699 


5837 


5976 


6114 6253 6391 


9 


8 


6663 


6806 


6945 


7083 


7221 


7360 


7493 7636 7775 


9 


9 


8052 


8190 


8328 


8467 


8605 


8743 


88S2 9020 9158 


9 


3140 


9435 


9573 


9711 


9850 


9983 


/0126 
'1509 


4970265 0403 4970541 


138 


1 


4970818 


0956 


4971094 


1232 


4971371 


1647 1785 1924 


8 


2 


2200 


2338 


2476 


2615 


2753 


2391 


3029 3167 3306 


8 


3 


3582 


3720 


3853 


3996 


4135 


4273 


4411 4549 4687 


8 


4 


4964 


5102 


5240 


5378 


5516 


5654 


5792 5930 


' 6068» 8 li 


5 


6345 


6483 


6621 


6759 


6897 


7035 


7173 7311 


! 7449 


8 i 


6 


7725 


7863 


8001 


8139 


8277 


8415 


8553 8691 


8829 


8 1 


7 


9105 


9243 


9381 


9519 


9657 


9795 


9933 /0071 
4981313 ' 1451 


4980209 


8 


8 


4980485 


0623 


4980761 


0899 


4981037 


1175 


1589 


8 


9 


1865 


2002 


2140 


2278 


2416 


2554 


2692 2830 


2968 


8 


3150 


3243 


3381 


3519 


3657 


3795 


3933 


4071 4203 


4346 


8 


1 


4622 


4760 


4897 


5035 


5173 


5311 


5449 5587 


5724 


8 


2 


6000 


6138 


6275 


6413 


6551 


6689 


63-26 6964 


7102 


8 


3 


7377 


7515 


7653 


7791 


7928 


8066 


8204 8341 


8479 


8 


4 


8755 


8892 


9030 


9168 


9305 


9443 


9531 9713^ 9856 


8 


5 


4990131 


0269 


4990407 


0544 


4990682 


0819 


4990957 109514991232 
2333 247 1! 2608 


8 


6 


1508 


1645 


1783 


1920 


2053 


2196 


8 


7 


2683 


3021 


3158 


3296 


3434 


3571 


3709 3846 


3984 


8 


8 


4259 


4396 


4534 


4671 


4809 


4946 


5084 5221 


5359 


8 


9 


5634 


5771 


5909 


6046 


6184 


6321 


6459 6596 


6733 


8 


3160 


7008 


7146 


7283 


7421 


7553 


7695 


7833 7970 


8108 


8 


1 


8382 


8520 


8657 


6794 


8932 


9069 


9207 9344 


9431 


8 


2 


9756 


9393 


5000031 


0168 


5000305 


0443 


5000580 071785000855 
1953 2090 2227 
3325 3463 « 3600 


8 


3 


5001129 


1267 


1404 


1541 


1678 


1816 


8 


4 


2502 


2639 


2777 


2914 


3051 


3138 


8 


5 


3874 


4012 


4149 


4286 


4423 


4560 


4698 4835 4972] 


8 


6 


5246 


5333 


5521 


5658 


5795 


5932 


6069 6206 


6344 


8 


7 


6618 


6755 


6892 


7029 


7166 


7303 


7440 7578 


7715 


8 


8 


7989 


8126 


8263 


8400 


8537 


8674 


8311 8948 


9085 


8 


9 


9359 


9496 


9634 


9771 


9903 


.0045 
1415 


5010182 0319 


5010456 


137 


3170 


5010730 


0867 


5011004 


1141 


5011278 


1552 1688 


1825 




1 


2099 


2236 


2373 


2510 


2647 


2784 


2921 3058 


3195 




2 


3469 


3606 


3743 


3879 


4016 


4153 


4290 4427 


4564 




3 


4838 


4974 


5111 


5243 


5385 


5522 


5659 5796 


5932 




4 


6206 


6343 


6480 


6617 


6753 


6890 


7027 7164 


7301 




5 


7574 


7711 


7848 


7984 


8121 


8253 


8395 85311 8668 
9762 9399 5020035 




6 


8942 


9078 


9215 


9352 


9489 


9625 




7 


5020309 


0446 


5020582 


0719 


5020856 


0992 


5021129 1266 1402 




8 


1676 


1812 


1949 


2086 


2222 


2359 


2495 2632 2769 




9 


3042 


3178 


3315 


3452 


3588 


3725 


3861 3998 4i35 
T 8 9 






1 


2 


3 


4 


5 


6 


- . 



Table i.] logarithms of numbers from 1 to 36,000. 55 


Betweea 31800 = log.""! 4-5024271, and 32400 = log.~i 4-5105450. 


teixs 


1 3 


3 4: 


5 6 I T 8 


9 


dif. 


31SC 


5024408 4544 


5024681 4817 


5024954 5091 


5025227 5364 


5025500 


137 


1 


5773 5910 


6046 6183 


6319 6456 


6592 6729 


6865 


7 




7133 7-275 


7411 7548 


7684 7821 


7957 8093 


8230 


7 


3 


8503 8639 


8776 8912 


9049 9185 


9321 9458 


9594 


7 


4 


9367 /0003 
5031231 / 1367 


5030140 0276 


5030413 0549 


5030685 0822 


5030958' 7 II 


5 


1503 1640 


1776 1912 


2049 2185 


2321 


7 


6 


2594 2730 


2867 3003 


3139 3276 


3412 3548 


3684 


7 


7 


3957 4093 


4229 4366 


4502 4638 


4774 4911 


5047 


7 


8 


5319 5456 


5592 5728 


5864 6000 


6137 6273 


6409 


7 


9 


6631 6818 


6954 7090 


7-2-26 7362 


7498 7635 


7771 


7 


3190 


8043 8179 


8315 8451 


8587 8724 


8860 8996 


9132 


7 


1 


9404 9540 


9676 9812 


9948 /0085 
5041309 / 1445 


5040221 0357 


5040493 


7 


2 


5040765 0901 


5041037 1173 


1581 1717 


1853 


7 


3 


2125 2261 


2397 2533 


2669 2805 


2941 ,3077 


3213 


7 


4 


3485 3621 


3757 3893 


4029 4165 


4301 4437 


4573 


136 


5 


4845 4980 


5116 5252 


5388 5524 


5660 5796 


5932 


6 


6 


6204 6339 


6475 6611 


6747 6883 


7019 7155 


7291 


6 


7 


7562 7698 


7834 7970 


8106 8241 


8377 8513 


8649 


6 


8 


8920 9056 


9192 9328 


9464 9599 


9735 9871 


5050007 


6 


9 


5050278 0414 


5050550 0685 


5050821 0957 


5051093 1228 


1364 


6 


3200 


1635 1771 


1907 2043 


2178 2314 


2450 2585 


2721 


6 


1 


2992 3128 


3264 3399 


3535 3671 


3606 3942 


4078 


6 


2 


4349 4485 


4620 4756 


4891 5027 


5163 5298 


5434 


6 


3 


5705 5841 


5976 6112 


6247 6383 


6518 6654 


6790 


6 


4 


7061 7196 


7332 7467' 7603 7738' 7874 8009' 8145' 6 \\ 


5 


8416 8551 


8687 8822 


8953 9093 


- 9229 9364 


9500 


6 


6 


9771 9906 


5060042 0177 


5060312 0448 


5060583 0719 


5060854 


6 


7 


5061125 1260 


1396 1531 


1667 1802 


1937 2073 


2208 


6 


8 


2479 2614 


2750 2885 


3020 3156 


3-291 3428 


3562 


6 


9 


3833 3968 


4103 4238 


4374 4509 


4644 4780 


4915 


6 


3210 


5186 5321 


5456 5591 


5727 5862 


5997 6133 


6268 


6 


1 


6533 6674 


6809 6944 


7079 7214 


7350 7485 


7620 


6 


2 


7891 8026 


8161 8296 


8431 8567 


8702 8837 


8972 


6 


3 


9242 9378 


9513 9648 


9783 9918 5070053 01881 


50703-24 


6 


4 


5070594 0729 


5070864 0999 


5071134 1269 1405 1540' 1675' 6 || 


5 


1945 2080 


2215 2350 


2485 2620 


2755 2890 


3025 


6 


6 


3295 3430 


3566 3701 


3836 3971 


4106 4241 


4376 


6 


7 


4646 4781 


4916 5051 


5186 5321 


5456 5590 


5725 


6 


8 


5995 6130 


6265 6400 


6535 6670 


6805 6940 


7075 


6 


9 


7345 7480 


7614 7749 


7884 8019 


8154 8289 


84-24 


6 


3220 


8694 8828 


8963 9098 


9233 9368 


9503 9638 


9772 


135 


1 


5080042 0177 


5080312 0447 


5080581 0716 


5080851 0986 


5081121 


5 


2 


1390 1525 


1660 1794 


1929 2054 


2199 2334 


2468 


5 


3 


2738 2873 


3007 3142 


3277 3411 


3546 3681 


3816 


5 


4 


4085 4220 


4354 4489 


4624 4758 


4893 5028 


5163 


5 


5 


5432 5567 5701 5836 


5970 6105 


6240 6374 


mm 


5 


6 


6778 6913 7047 7182 


7317 7451 


7586 7720 


7855 


5 


7 


8124 8259 8393 8528 


8663 8797 


8932 9066 


9201 


5 


8 


9470 9604 9739 9873 


5090008 0142 


5090277 0411 


5090546 


5 


9 


5090815 0949 


5091084 1218 


1353 1487 


1622 1756 


1891 


5 


3230 


2160 2294 


2429 2563 


2697 2832 


2966 3101 


3235 


5 1 


1 


3504 3638 


3773 3907 


4042 4176 


4310 4445 


Abl^ 


5 1 


2 


4848 4982 


5117 5251 


5385 5520 


5654 5788 


5923 


5 1 

5 1 


3 


6191 63-26 


6460 6594 


6729 6863 


6997 7132 


7266 


4 


7534 7669 


7803 7937 


8072 8206 


8340 8474 86091 


5 1 


5» 8877 9011 


9146 9280 


9414 9548 9682 9817" 9951! 5 || 


6 5100219 0354 


5100488 0622 


5100756 0890 


5101024 1159 5101293 5 || 


7 1561 1695 


1829 1964 


2098 2232 


2366 2500 2634 


5 1 


! 8 2903 3037 


3171 3305 


3439 3573 


3707 3841 3975 


5 


9| 4244 4378 


4512 4646 


4780 4914 


5048 5182 5316 


134 


111 2 1 


3 4 


5 6 T 8 f 9 1 











""^ -t 




56 LOGAKITHMS OF NUMBERS FEOM 1 TO 36,000. [Table I. \ 




Between 32400 = log." 


"1 4-5105450, and 33000 = log ."' 4.5185139. 




tens. 


1 


2 


3 


4 


5 


6 1 


T 8 1 


9 


dif. 




3240 


5105584 


5718 


5105852 


5986 


5106120 


62541 


5106388 6522 


5106656 


134 




1 


6924 


7058 


7192 


7326 


7460 


7594 


7728 7862 


7996 


4 




2 


8264 


8398 


8532 


8666 


8800 


8934 


9068 9202 


9336 


4 




3 


9603 


9737 


9871 


/0005 


5110139 


0273 


6110407 0541 


5110675 


4 




4 5110942 


1076 5111210/ 1344' 


1478 


1612' 


1745 1879 


2013 


4 




5 


2281 


2415 


2548 


2682 


2816 


2950 


3084 3218 


3351 


4 




6 


3619 


3753 


3887 


4020 


4154 


4288 


4422 4555 


4689 


4 




7 


4957 


5090 


5224 


5358 


5492 


5625 


5759 5893 


6026 


4 




8 


6294 


6428 


6561 


6695 


6829 


6962 


7096 7230 


7363 


4 




9 


7631 


7764 


7898 


8032 


8165 


8299 


8433 8566 


8700 


4 




3250 


8967 


9101 


9234 


9368 


9502 


9635 


9769 9903 


5120036 


4 




1 


5120303 


0437 


5120570 


0704 


5120838 


0971 


5121105 1238 


1372 


4 




2 


1639 


1772 


1906 


2040 


2173 


2307 


2440 2574 


2707 


4 




3 


2974 


3108 


3241 


3375 


3508 


3642 


3775 3909 


4042 


4 




4 


4309 


4443 


4576 


4709 


4843 


4976 


5110 5243 


5377 


4 




5 


5643 


5777 


5910 


6044 


6177 


6310 


6444 6577 


6711 


4 




6 


6977 


7111 


7244 


7377 


7511 


7644 


7778 7911 


8044 


4 




7 


8311 


8444 


8578 


8711 


8844 


8978 


9111 9244 


9377 


4 




8 


9644 


9777 


9911 


/0044 


5130177 


0311 


5130444 0577 


5130710 


4 




9 


5130977 


1110 


5131243 / 1377 


1510 


1643 


1776 1910 


2043 


4 




3260 


2309 


2442 


2576 


2709 


2842 


2975 


3108 3242 


3375 


4 




1 


3641 


3774 


3908 


4041 


4174 


4307 


4440 4573 


4706 


4 




2 


4973 


5106 


5239 


5372 


5505 


5638 


5771 5905 


6038 


133 




3 


6304 


6437 


6570 


6703 


6836 


6969 


7102 7235 


7368 


3 




4 


7635 


7768 7901 


8034 


8167 


8300 


8433 8566 


8699 


3 




5 


8965 


9098 


9231 


9364 


9497 


9630 


9763 9896 


5140029 


3 




6 


5140295 


0428 


5140561 


0694 


5140827 


0960 


5141093 1225 


1358 


3 




7 


1624 


1757 


1890 


2023 


2156 


2289 


2422 2555 


2688 


3 




8 


2953 


3086 


3219 


3352 


3485 


3618 


3751 3883 


4016 


3 




9 


4282 


4415 


4548 


4681 


4813 


4946 


5079 5212 


5345 


3 




3270 


5610 


5743 


5876 


6009 


6142 


6274 


6407 6540 


6673 


3 




1 


6938 


7071 


7204 


7336 


7469 


7602 


7735 7867 


8000 


3 




2 


8266 


8398 


8531 


8664 


8797 


8929 


9062 9195 


9327 


3 




3 


9593 


9725 


9858 


9991 


5150123 


0256 


5150389 0521 


5150654 


3 




4 


5150919 


1052 


5151185 


1317 


1450 


1583 


1715 1848 


1980 


3 




5 


2246 


2378 


2511 


2643 


2776 


2909 


3041 3174 


3306 


3 




6 


3571 


3704 


3837 


3969 


4102 


4234 


4367 4499 


4632 


3 




7 


4897 


5029 


5162 


5294 


5427 


5560 


5692 5825 


5957 


3 




8 


6222 


6354 


6487 


6619 


6752 


6884 


7017 7149 


7282 


3 




9 


7547 


7679 


7811 


7944 


8076 


8209 


8341 8474 


8606 


3 




3280 


8871 


9003 


9136 


9268 


9400 


9533 


9665 9798 


9930 


3 




1 


5160195 


0327 


5160459 


0592 


5160724 


0856 


5160989 1121 


5161253 


3 




2 


1518 


1650 


1783 


1915 


2047 


2180 


2312 2444 


2577 


3 




3 


2841 


2973 


3106 


3238 


3370 


3502 


3635 3767 


3899 


3 




4 


4164 


4296 


4428 


4560 


4693 


4825 


4957 5089" 5222 


3 




5 


5486 


5618 


5750 


5883 


6015 


6147 


6279 6411 


6543 


3 




6 


6808 


6940 


7072 


7204 


7336 


7469 


7601 7733 


7865 


132 




7 


8129 


8261 


8393 


8526 


8658 


8790 


8922 9054 


9186 


2 




8 


9450 


9582 


9714 


9846 


9978 


/Olll 
'1431 


5170243 0375 


5170507 


2 




9 


5170771 


0903 


5171035 


1167 


5171299 


1563 1695 


1827 


2 




3290 


2091 


2223 


2355 


2487 


2619 


2751 


2883 3015 


3147 


2 




1 


3411 


3543 


3675 


3807 


3939 


4071 


4202 4334 


4466 


2 




2 


4730 


4862 


4994 


5126 


5258 


5390 


5522 5654 


5785 


2 




3 


6049 


6181 


6313 


6445 


6577 


6709 


6840 6972 


7104 


2 




4 


7368 


7500 


7631 


7763 


7895 


8027 


8159 8291 


8422 


2 




5 


8686 


8818 


8950 


9081 9213 


9345 


9477 9608 f 9740 


2 




6 


5180004 


0136 


5180267 


0399 


5180531 


0663 


5180794 0926 


5181058 


2 




7 


1321 


1453 


1585 


1716 


1848 


1980 


2111 2243 


2375 


2 




8 


2638 


2770 


2902 


3033 


3165 


3297 


3428 3560 


3692 


2 




9 


3955 


4086 


4218 


4350 


4481 


4613 


4745 4876 


5008 


2 




^^^ 


1 


2 


3 


4 


5 


6 


7 8 


9 


^__, 


• 



Table I.] 


LOGARITHMS 


OF NUMBERS 


FROM 1 TO 36,000. 57 




Between 33000 = log." 


' 4-5185139, and 33600 ^ log 


."^•5263393. 1 




tens. 


1 


2 


3 


4 


5 


6 T 


8 


9 dif. 1 




3300 


5185271 


5403 


5185534 


5666 


5185797 


5929 


5186061 


6192 


5186324 


132 




1 


6587 


6718 


6850 


6981 


7113 


7245 


7376 


7508 


7639 


2 




2 


7902 


8034 


8165 


8297 


8428 


8560 


8691 


8823 


8954 


2 




3 


9217 


9349 


9480 


9612 


9743 


9875 


5190006 


0137 


5190269 


2 




4 


5190532 


0663 


5190795 


0926 


5191058 


1189 


1320 


1452 


1583 


2 




5 


1846 


1977 


2109 


2240 


2372 


2503 


2634 


2766 


2897 


2 




6 


3160 


3291 


3423 


3554 


3685 


3817 


3948 


4079 


4211 


2 




7 


4473 


4605 


4736 


4867 


4999 


5130 


5261 


5392 


5524 


2 




8 


5786 


5918 


6049 


6180 


6311 


6443 


6574 


6705 


6836 


2 




9 


7099 


7230 


7361 


7493 


7624 


7755 


7886 


8018 


8149 


2 




'3310 


8411 


8542 


8674 


8805 


8936 


9067 


9198 


9329 


9461 


2 




1 


9723 


9854 


9985 


/0116 
'1428 


5200248 


0379 


5200510 


0641 


5200772 


131 




2 


5201034 


1166 


5201297 


1559 


1690 


1821 


1952 


2083 






3 


2345 


2477 


2608 


2739 


2870 


3001 


3132 


3263 


3394 






4 


3656 


3787 


3918 


4049 


4180 


4311 


4442 


4573 


4704 






5 


4966 


5097 


5228 


5359 


5490 


5621 1 5752 


5883 


6014 






6 


6276 


6407 


6538 


6669 


6800 


6931 


7062 


7193 


7324 






7 


7586 


7717 


7847 


7978 


8109 


8240 


8371 


8502 


8033 






8 


8895 


9026 


9156 


9287 


9418 


9549 


9680 


9811 


9942 






9 


5210203 


0334 


5210465 


0596 


5210727 


0858 


5210988 


1119 


5211250 






3320 


1512 


1642 


1773 


1904 


2035 


2166 


2296 


2427 


2558 






1 


2820 


2950 


3081 


3212 


3343 


3473 


3604 


3735 


3866 






2 


4127 


4258 


4388 


4519 


4650 


4781 


4911 


5042 


5173 






3 


5434 


5565 


5695 


5826 


5957 


6088 


6218 


6349 


6479 






4 


6741 


6871 


7002 


7133 


7263 


7394 


7525 


7655 


7786 






5 


6047 


8178 


I 8308 


8439 


8570 


8700 


8831 


8961 


9092 






6 


9353 


9484 


9614 


9745 


9875 


/0006 
'1311 


5220136 


0267 


5220397 






7 


5220659 


0789 


5220920 


1050 


5221181 


1442 


1572 


1703 






8 


1964 


2094 


2225 


2355 


.2486 


2616 


2747 


2877 


3007 






9 


3268 


3399 


3529 


3660 


3790 


3921 


4051 


4181 


4312 


130 




3330 


4573 


4703 


4834 


4964 


5094 


5225 


5355 


5486 


5616 







1 


5877 


6007 


6137 


6268 


6398 


6529 


6659 


6789 


6920 







2 


7180 


7311 


7441 


7571 


7702 


7832 


7962 


8093 


8223 





• 


3 


8483 


8614 


8744 


8874 


9005 


9135 


9265 


9395 


9526 







4 


9786 


9916 


5230047 


0177 


5230307 


0437 


5230568 


0698 


5230828 







5 


5231089 


1219 


1349 


1479 


1609 


1740 


1870 


2000 


2130 







6 


2391 


2521 


2651 


2781 


2911 


3041 


3172 


3302 


3432 







7 


3692 


3822 


3952 


4083 


4213 


4343 


4473 


4603 


4733 







8 


4993 


5124 


5254 


5384 


5514 


5644 


5774 


5904 


6034 







9 


6294 


6424 


6554 


6684 


6814 


6945 


7075 


7205 


7335 







3340 


7595 


7725 


7855 


7985 


8115 


8245 


8375 


8505 


6635 


/O 




1 


8895 


9025 


9155 


9285 


9415 


9545 


9675 


9805 


9935 







2 


5240194 


0324 


5240454 


0584 


5240714 


0844 


5240974 


1104 


5241234 







3 


1494 


1624 


1753 


1883 


2013 


2143 


2273 


2403 


2533 







4 


2793 


2922 


3052 


3182 


3312 


3442 


3572 


3702 


3831 







5 


4091 


4221 


4351 


4481 


4610 


4740 


4870 


5000 


5130 







6 


5389 


5519 


5649 


5779 


5908 


6038 


6168 


6298 


6427 







7 


6687 


6817 


6946 


7076 


7206 


7336 


7465 


7595 


■ 7725 







8 


7984 


8114 


8244 


8373 


8503 


8633 


8762 


8892 


9022 







9 


9281 


9411 


9540 


9670 


9800 


9929 


5250059 


0189 


5250318 







3350 


5250578 


0707 


5250837 


0967 


5251096 


1226 


1355 


1485 


1615 







1 


1874 


2003 


2133 


2263 


2392 


2522 


2651 


2781 


2911 







2 


3170 


3299 


3429 


3558 


3688 


3817 


3947 


4076 


4206 







3 


4465 


4595 


4724 


4854 


4983 


5113 


5242 


5372 


5501 







4 


5760 


5890 


6019 


6148 


6278 


6407 


6537 


6666 


6796 







5 


7055 


7184 


7314 


7443 


7572 


7702 


7831 


7961 


8090 







6 


8349 


8478 


8608 


8737 


8867 


8996 


9125 


9255 


9384 







7 


9643 


9772 


9902 


^0031 


5260160 


0290 


5260419 


0548 


5260678 







8 


5260936 


1066 


5261195/ 13241 


1454 


1583 


1712 


1841 


1971 







9 


2229 


2359 


2488 


2617 


2746 


2876 


3005 


3134 


3264 









1 


2 


3 


4 5 


6 


T 


8 


9 






^■~' 










H- 












1 



1 58 LOGARITHMS OF NUMBERS FROM 1 


TO 36,000. 


[Table I. 


Between 


33600 = log.' 


"1 4-5263393, and 34200 = log.~i 45340261. | 


tens. 


1 


2 


3 


4 


5 


6 


T 


8 


9 


dif. 


3360 


5263522 


3651 


5263781 


3910 


5264039 


4168 


5264297 


4427 


5264556 


130 


1 


4814 


4944 


5073 


5202 


5331 


5460 


5590 


5719 


5848 





2 


6106 


6235 


6365 


6494 


6623 


6752 


6881 


7010 


7140 





3 


7398 


7527 


7656 


7785 


7914 


8043 


8173 


8302 


8431 





4 


8689 


8818 


8947 


9076 


9205 


9334 


9463 


9593 


9722 


129 


5 


99S0 


/0109 
'1399 


5270238 


0367 


5270496 


0625 


5270754 


0883 


5271012 


9 ! 


6 


5271270 


1528 


1657 


1786 


1915 


2044 


2173 


2302 


9 i 


7 


2560 


2689 


2818 


2947 


3076 


3205 


3334 


3463 


3592 


9 


8 


3850 


3979 


4108 


4237 


4366 


4494 


4623 


4752 


4881 


9 


9 


5139 


5268 


5397 


5526 


5655 


5783 


5912 


6041 


6170 


9! 


3370 


6428 


6557 


6686 


6814 


6943 


7072 


7201 


7330 


7459 


9 ' 


1 


7716 


7845 


7974 


8103 


8232 


8360 


8489 


8618 


8747 


9| 


2 


9004 


9133 


9262 


9391 


9520 


9648 


9777 


9906 


5280035 


9 


3 


5280292 


0421 


5280550 


0678 


5280807 


0936 


5281065 


1193 


1322 


9 


4 


1579 


1708 


1837 


1966 


2094 


2223 


2352 


2480 


2609 


9 


5 


2866 


2995 


3124 


3252 


3381 


3510 


3638 


3767 


3896 


9 


6 


4153 


4282 


4410 


4539 


4668 


4796 


4925 


5053 


5182 


9 


7 


5439 


5568 


5696 


5825 


5954 


6082 


6211 


6339 


6468 


9 1 


8 


6725 


6854 


6982 


7111 


7239 


7368 


7496 


7625 


7753 


9 1 


1 9 


8010 


8139 


8267 


8396 


8525 


8653 


8782 


8910 


9039 


9! 


•3380 


9295 


9424 


9552 


9681 


9809 


9938 


5290066 


0195 


5290323 


9, 


1 


5290580 


0709 


5290837 


0965 


5291094 


1222 


1351 


1479 


1608 


9 


2 


1864 


1993 


2121 


2250 


2378 


2506 


2635 


2763 


2892 


9 


3 


3148 


3277 


3405 


3533 


3662 


3790 


3919 


4047 


4175 


9 


4 


4432 


4560 


4689 


4817 


4945 


5074 


5202 


5330 


5458 


9 1 


5 


5715 


5843 


5972 


6100 


6228 


6356 


6485 


6613 


6741 


9 


6 


6998 


7126 


7254 


7383 


7511 


7639 


7767 


7896 


8024 


9 


7 


8280 


8408 


8537 


8665 


8793 


8921 


9049 


9178 


9306 


9 


8 


9562 


9690 


9819 


9947 


5300075 


0203 


5300331 


0459 


5300583 


9 


9 


5300844 


0972 


5301100 


1228 


1356 


1485 


1613 


1741 


1369 


9 


3390 


2125 


2253 


2381 


2509 


2637 


2766 


2894 


3022 


3150 


9 


1 


3406 


3534 


3662 


3790 


3918 


4046 


4174 


4302 


4430 


.9 


2 


4686 


4814 


4943 


5071 


5199 


5327 


5455 


5583 


5711 


12s 


3 


5967 


6095 


6223 


6351 


6479 


6607 


6734 


6862 


6990 


8 


4 


7246 


7374 


7502 


7630 


7758 


7886 


8014 


8142 


8270 


8 


5 


8525 


8654 


8782 


8909 


[ 9037 


9165 


9293 


9421 


9549 


8 


6 


9805 


9933 


5310060 


0188 


5310316 


0444 


5310572 


0700 


5310828 


8 


7 


5311083 


1211 


1339 


1467 


1595 


1722 


1850 


1978 


2106 


8 


8 


2362 


2489 


2617 


2745 


2873 


3001 


3128 


3256 


3384 


8 


9 


3639 


3767 


3895 


4023 


4150 


4278 


4406 


4534 


4661 


8 


3400 


4917 


5045 


5172 


5300 


5428 


5556 


5683 


5811 


5939 


8 


1 


6194 


6322 


6449 


6577 


6705 


6832 


6960 


7088 


7215 


8 


2 


7471 


7598 


7726 


7854 


7981 


8109 


8237 


8364 


8492 


8 


S 


8747 


8875 


9002 


9130 


9258 


9385 


9513 


9640 


9768 


8 


4 


5320023 


0151 


5320278 


0406 


5320533 


0661 


5320789 


0916 


5321044 


8 


5 


1299 


1426 


1554 


1681 


1809 


1936 


2064 


2191 


2319 


8 


6 


2574 


2701 


2829 


2956 


3034 


3211 


3339 


3466 


3594 


8 


7 


3849 


3976 


4104 


4231 


4359 


4486 


4614 


4741 


4868 


8 1 


8 


5123 


5251 


5378 


5506 


5633 


5760 


5888 


6015 


6143 


8 1 


9 


6397 


6525 


6652 


6780 


6907 


7034 


7162 


7289 


7416 


8 


3410 


7671 


7799 


7926 


8053 


8181 


8308 


8435 


8563 


8690 


8 


1 


8945 


9072 


9199 


9326 


9454 


9581 


9708 


9836 


9963 


8 


2 


5330218 


0345 


5330472 


0599 


5330727 


0854 


5330981 


1108 


5331236 


8 


3 


1490 


1617 


1745 


1872 


1999 


2126 


2254 


2381 


2508 


8 


4 


2762 


2890 


3017 


3144 


3271 


3398 


3526 


3653 


3780 


8 


5 


4034 


4161 


4289 


4416 


4543 


4670 


4797 


4924 


5051 


8 


6 


5306 


5433 


5560 


5687 


5814 


5941 


6068 


6196 


6323 


8 


7 


6577 


6704 


6831 


6958 


7085 


7212 


7339 


7466 


7594 


8 


8 


7848 


7975 


8102 


8229 


8356 


8483 


8610 


8737 


8864 


8 


9 


9118 


9245 


9372 


9499 


9626 


9753 


9880 


0007 


5340134 


127 


— - 


1 


2 


3 


4 


5 


6 


7 


8 


9 


1 



1 Tabic I.] 


LOGARITHMS 


OF NUMBERS 


FROM 


1 TO 36,000. 


59 1 


! Between 34200 


= log. 


' 4-5340261, and 34800 = log "i 4-5415792. || 


\tens. 


1 


z 


3 


4 


5 


(> 


7 8 


9 


dif. 


3420 


5340383 


0515 


5340642 


0769 


5340896 


1023 


5341150 1277 


5341404 


127 


1 


1658 


1785 


1912 


2039 


2165 


2292 


2419 2546 


2673 




2 


2927 


3054 


3181 


3308 


3435 


3561 


3688 3815 


3942 




3 


4196 


4323 


4450 


4576 


4703 


4830 


4957 5084 


5211 




4 


5464 


5591 


5718 


5845 


5972 


6099 


6225 6352 


6479 




5 


6733 


6859 


6986 


7113 


7240 


7366 


7493 7620 


7747 




6 


8000 


8127 


8254 


8381 


8507 


8634 


8761 8888 


9014 




7 


9268 


9394 


9521 


9648 


9775 


9901 


5350028 0155 


5350281 




8 


5350535 


0662 


5350788 


0915 


5351042 


1168 


1295 1422 


1548 




9 


1802 


1928 


2055 


2181 


2308 


2435 


2561 2688 


2815 




3430 


3068 


3194 


3321 


3448 


3574 


3701 


3827 3954 


4081 




1 


4334 


4460 


4587 


4713 


4840 


4967 


5093 5220 


5346 




2 


5599 


5726 


5852 


5979 


6105 


6232 


6359 6485 


6612 




3 


6865 


6991 


7118 


7244 


7371 


7497 


7623 7750 


7876 




4 


8129 


8256 


8382 


8509 


8635 


8762 


8888 9015 


9141 




5 


9394 


9520 


9647 


9773 


9900 


/ 0026 
'1290 


5360152 0279 


5360405 




6 


5360658 


0784 


5360911 


1037 


5361163 


1416 1543 


1669 




7 


1922 


2048 


2174 


2301 


2427 


2553 


2680 2806 


2932 




8 


3185 


3311 


3438 


3564 


3690 


3817 


3943 4069 


4195 




9 


4443 


4574 


4701 


4827 


4953 


5079 


5206 5332 


5458 




3440 


5711 


5837 


5963 


6089 


6216 


6342 


6468 6594 


6721 




1 


6973 


7099 


7225 


7352 


7478 


7604 


7730 7856 


7982 




2 


8235 


8361 


8487 


8613 


8739 


8866 


8992 9118 


9244 




3 


9496 


9622 


9749 


9875 


5370001 


0127 


5370253 0379 


5370505 




4 


5370758 


0884 


5371010 


1136 


1262 


1388 


1514 1640 


1766 




5 


2018 


2144 


2270 


2396 


2523 


2649 


2775 2901 


3027 




6 


3279 


3405 


3531 


3657 


3783 


3909 


4035 4161 


4287 




7 


4539 


4865 


4791 


4917 


5043 


5169 


5295 5421 


5547 


126 


8 


5799 


5924 


6050 


6176 


6302 


6428 


6554 6680 


6806 


6 


9 


7058 


7184 


7310 


7436 


7561 


7687 


7813 7939 


8065 


6 


3450 


8317 


8443 


8569 


8694 


8820 


8946 


9072 9198 


9324 


6 


1 


9575 


6701 


9827 


9953 


5380079 


0205 


5380330 0456 


5380582 


6 


2 


5380834 


0959 


5381085 


1211 


1337 


1463 


1588 1714 


1840 


6 


3 


2092 


2217 


2343 


2469 


2595 


2720 


2846 2972 


3098 


6 


4 


3349 


3475 


3601 


3726 


3852 


3978 


4103 4229 


4355 


6 


5 


4606 


4732 


4858 


4983 


5109 


5235 


5360 5486 


5612 


6 


6 


5863 


5989 


6114 


6240 


6366 


6491 


6617 6743 


6668 


6 


7 


7119 


7245 


7371 


7496 


7622 


7747 


7873 7999 


8124 


6 


8 


8375 


8501 


8627 


8752 


8878 


9003 


9129 9255 


9380 


6 


9 


9631 


9757 


9882 


.0008 
'1263 


5390133 


0259 


5390384 0510 


5390635 


6 


3460 


5390887 


1012 


5391138 


1389 


1514 


1640 1765 


1891 


6 


1 


2141 


2267 


2392 


2518 


2643 


2769 


2894 3020 


3145 


6 


2 


3396 


3522 


3647 


3772 


3898 


4023 


4149 4274 


4400 


6 


3 


4650 


4776 


4901 


5027 


5152 


5277 


5403 5528 


5653 


6 


4 


5904 


6030 


6155 


6280 


6406 


6531 


6656 6782 


6907 


6 


5 


7158 


7283 


7408 


7534 


7659 


7784 


7910 8035 


8160 


6 


6 


8411 


8536 


8661 


8787 


8912 


9037 


9163 9288 


9413 


125 


7 


9664 


9789 


9914 


/0039 
'1292 


5400165 


0290 


5400415 0540 


5400666 


5 


8 


5400916 


1041 


5401167^ 


1417 


1542 


1667 1793 


1918 


5 


9 


2168 


2293 


2419 


2544 


2669 


2794 


2919 3044 


3170 


5 


3470 


3420 


3545 


3670 


3795 


3920 


4046 


4171 4296 


4421 


5 


1 


4671 


4796 


4921 


5047 


5172 


5297 


5422 5547 


5672 


B 


2 


,5922 


6047 


6172 


6297 


6423 


6548 


6673 6798 


6923 


5 


3 


7173 


7298 


7423 


7548 


7673 


7798 


7923 8048 


8173 


5 


4 


8423 


8548 


8673 


8798 


8923 


9048 


9173 9298 


9423 


5 


5 


9673 


9798 


9923 


/0048 
'1297 


5410173 


0298 


5410423 0548 


5410673 


5 


6 


5410923 


1048 


5411172^ 


1422 


1547 


1672 1797 


1922 


5 


7 


2172 


2297 


2422 


2546 


2671 


2796 


2921 3046 


3171 


5 


8 


3421 


3546 


3670 


3795 


3920 


4045 


4170 4295 


4419 


5 


9 


4669 


4794 


4919 


5044 


5168 


5293 


5418 5543 


5668 


5 


=as 


1 


2 


3 


4r 


S 


6 


T 8 













1 


■60 LOGARITHMS OF NUMBERS FROM 1 TO 36,000. [Tobk I. {| 


Between 34800 = log." 


' 4-5415792, and 35400 = log 


ri 4-5490033. {| 


\tens. 1 


2 3 


4: 


S 


6 


T 


8 


9 


dif. 


34.80 


5415917 


6042 5416167 


6292 


5416416 


6541 


5416666 


6791 


5416915 


125 


1 


7165 


7290 7415 


7539 


7664 


7789 


7913 


8038 


8163 


5 


2 


8412 


8537 8662 


8787 


8911 


9036 


9161 


9285 


9410 


5 


3 


9659 


9784 9909 


,0033 

'1280 


5420158 


0283 


5420407 


0532 


5420657 


5 


4 


5420906 


1031 5421155^ 


1404 


1529 


1654 


1779 


1903 


5 


5 


2152 


2277 


2402 


2526 


2651 


2775 


2900 


3025 


3149 


5 


6 


3398 


3523 


3648 


3772 


3897 


4021 


4146 


4270 


4395 


5 


7 


4644 


4769 


4893 


5018 


5142 


5267 


5391 


5516 


5640 


5 


8 


5889 


6014 


6138 


6263 


6387 


6512 


6636 


6761 


6885 


5 


9 


7134 


7259 


7383 


7508 


7632 


7756 


7881 


8005 


8130 


5 


3490 


8379 


8503 


8628 


8752 


8876 


9001 


9125 


9250 


9374 


5 


1 


9623 


9747 


9872 


9996 


5430120 


0245 


5430369 


0494 


5430618 


b 


2 


5430867 


0991 


5431115 


1240 


1364 


1488 


1613 


1737 


1862 


B 


3 


2110 


2235 


2359 


2483 


2607 


2732 


2856 


2980 


3105 


5, 


4 


3353 


3478 3602 


3726 


3850 


3975 


4099 


4223 


4348 


5 


5 


4596 


4720 


4845 


4969 


5093 


5217 


5342 


5466 


5590 


S 


6 


5838 


5963 


6087 


6211 


6335 


6460 


6584 


6708 


6832 


5 


7 


7081 


7205 


7329 


7453 


7577 


7701 


7826 


7950 


8074 


b 


8 


8322 


8446 


8571 


8695 


8819 


8943 


9067 


9191 


9315 


5 


9 


9564 


9688 


9812 


9936 


5440060 


0184 


5440308 


0432 


5440556 


124 


3500 


5440805 


0929 


5441053 


1117 


1301 


1425 


1549 


1673 


1797 




1 


2045 


2169 


2293 


2417 


2541 


2665 


2789 


2913 


3037 




2 


3285 


3409 


3533 


3657 


3781 


3905 


4029 


4153 


4277 




3 


4525 


4649 


4773 


4897 


5021 


5145 


5269 


5393 


5517 




4 


5765 


5889 


6013 


6137 


6261 


6385 


6508 


6632 


6756 




5 


7004 


7128 


7252 


7376 


7500 


7624 


7747 


7871 


7995 




6 


8243 


8367 


8491 


8615 


8738 


8862 


8986 


9110 


9234 




7 


9481 


9605 


9729 


9853 


9977 


/OlOl 
'1339 


5450224 


0348 


5450472 




8 


5450720 


0843 


5450967 


1091 


5451215 > 


1462 


1586 


1710 




9 


1957 


2081 


2205 


2329 


2452 


2576 


2700 


2824 


2947 




3510 


3195 


3319 


3442 


3566 


3690 


3813 


3937 


4061 


4185 




1 


4432 


4556 


4679 


4803 


4927 


5050 


5174 


5298 


5421 




2 


5669 


5792 


5916 


6040 


6163, 


6287 


6411 


6534 


6658 




3 


6905 


7029 


7152 


7276 


• 7400 


7523 


7647 


7770 


7894 




4 


8141 


8265^ 8388 


8512 8635 


8759 


8883 


9006 


9130 




5 


9377 


9500 


9624 


9747 


9871 


9995 


5460118 


0242 


5460365 




6 


5460612 


0736 


5460859 


0983 


5461106 


1230 


1353 


1477 


1600 




7 


1847 


1971 


2094 


2218 


2341 


2465 


2588 


2711 


2835 




8 


3082 


3205 


3329 


3452 


3576 


3699 


3822 


3946 


4069 




9 


4316 


4439 


4563 


4686 


4810 


4933 


5056 


5180 


5303 




3520 


5550 


5673 


5797 


5920 


6043 


6167 


6290 


6414 


6537 




1 


6784 


6907 


7030 


7154 


7277 


7400 


7524 


7647 


7770 




2 


8017 


8140 


8263 


8387 


8510 


8633 


8757 


8880 


9003 




3 


9250 


9373 


9496 


9620 


9743 


9866 


9989 


/0113 
'1345 


5470236 




4 


5470482 


0605 


5470729 


0852 


5470975 


1098 


5471222 ' 


1468 




5 


1714 


1838 


1961 


2084 


2207 


2330 


2454 


2577 


2700 




6 


2946 


3069 


3193 


3316 


3439 


3562 


3685 


3808 


3931 


123 


7 


4178 


4301 


4424 


4547 


4670 


4793 


4916 


5040 


5163 




8 


5409 


5532 


5655 


5778 


5901 


6024 


6147 


6270 


6394 




9 


6640 


6763 


6886 


7009 


7132 


7255 


7378 


7501 


7624 




3530 


7870 


7993 


8116 


8239 


8362 


8485 


8608 


8731 


8854 


3 


1 


9100 


9223 


9346 


9469 


9592 


9715 


9838 


9961 


5480084 


3 


2 


5480330 


0453 


5480576 


0699 


5480822 


0945 


5481068 


1191 


1313 


3 


3 


1559 


1682 


1805 


1928 


2051 


2174 


2297 


2420 


2543 


a 


4 


2788 


2911^ 3034 


3157 


3280 


3403 


3526 


3648 


3771 


3 


5 


4017 


4140 


4263 


4386 


4508 


4631 


4754 


4877 


5000 


3 


6 


5245 


5368 


5491 


5614 


5737 


5859 


5982 


6105 


6228 


3 


7 


6473' 


6596 


6719 


6842 


6964 


7087 


7210 


7333 


7456 


3 


8 


7701 


7824 


7947 


8069 


8192 


8315 


8437 


8560 


8683 


3 


9 


6928 


9051 


9174 


9296 


9419 


9542 


9665 


9787 


9910 


3 




1 


2 1 3 


4 I S 


6 


T 


8 


9 















1 


Table i.] logarithms 


OF NUMBERS 


FROM 1 TO 36,000. 


61 


Between 35400 = log." 


~' 4-5490033, and 36000 = log ~i 4-5563025. | 


tens. 


1 2 


3 


4: 1 5 


6 


? 


8 


9 


\dif.\ 


3540 


5490155 0278 


5490401 


0523 


5490646 


0769 


5490891 


1014 


5491137 


123 


1 


1382 1505 


1627 


1750 


1872 


1995 


2118 


2240 


2363 


3 


2 


2608 2731 


2853 


2976 


3099 


3221 


3344 


3466 


3589 


3' 


3 


3S34 3957 


4079 


4202 


4324 


4447 


4569 


4692 


4815 


3i 


4 


5060 5182 


5305 


5427 


5550 


5672 


5795 


5917 


6040 


3 


5 


6285 6407 


6530 


6652 


6775 


6897 


7020 


7142 


7265 


3 


6 


7510 7632 


7755 


7877 


8000 


8122 


8245 


8367 


8489 


3 


7 


8734 8857 


8979 


9102 


9224 


9346 


9469 


9591 


9714 


3 


8 


9959 /0081 
5501182/1305 


5500203 


0326 


5500448 


0570 


5500693 


0815 


5500938 


3 


9 


1427 


1549 


1672 


1794 


1917 


2039 


2161 


3 


3550 


2406 2528 


2651 


2773 


2895 


3017 


3140 


3262 


3384 


3 


1 


3629 3751 


3874 


3996 


4118 


4240 


4363 


4485 


4607 


3 


2 


4852 4974 


5096 


5219 


5341 


5463 


5585 


5708 


5830 


3 


3 


6074 6197 


6319 


6441 


6563 


6685 


6808 


6930 


7052 


3 


4 


7296 7419 


7541 


7663 


7785 


7907 


8030 


8152 


8274 


3 


5 


8513 8640 


8763 


8885 


9007 


9129 


9251 


9373 


9495 


3 


6 


9740 9862 


9984 


/0106 
^1327 


5510228 


0350 


5510472 


0594 


5510717 


3 


7 


5510961 1083 


5511205^ 


1449 


1571 


1693 


1815 


1937 


3 


8 


2181 2304 


2426 


2548 


2670 


2792 


2914 


3036 


3158 


3 


9 


3402 3524 


3646 


3768 


3890 


4012 


4134 


4256 


4378 


3 


3560 


4622 4744 


4866 


4988 


5110 


5232 


5354 


5476 


5598 


122 


1 


5842 5964 


6086 


6208 


6329 


6451 


6573 


6695 


6817 


2 


2 


7061 7183 


7305 


7427 


7549 


7671 


7793 


7914 


6036 


2 


3 


8280 8402 


8524 


8646 


8768 


8890 


9011 


9133 


9255 


2 


4 


9499 9621 


9743 


9864 


9986 


/0108 5520230 


0352 


5520474 


2 


5 


5520717 0839 


5520961 


1083 


5521204 / 1326 


1448 


1570 


1692 


2 


6 


1935 2057 


2179 


2301 


2422 


2544 


2666 


2788 


2909 


2 


7 


3153 3275 


3396 


3518 


3640 


3762 


3883 


4005 


4127 


2 


8 


4370 4492 


4614 


4735 


4857 


4979 


5100 


5222 


5344 


2 


9 


5587 5709 


5831 


5952 


6074 


6196 


6317 


6439 


6561 


2 


3570 


6804 6925 


7047 


7169 


7290 


7412 


7534 


7655 


7777 


2 


1 


8020 8142 


8263 


8385 


8507 


8628 


8750 


8871 


8993 


2 


2 


9236 9358 


9479 


9601 


9722 


9844 


9965 


/0087 
'1302 


5530209 


2 


3 


5530452 0573 


5530695 


0816 


5530938 


1059 


5531181 


1424 


2 


4 


1667 1789 


1910 


2032 


2153 


2275 


2396 


2517 


2639 


% 


5 


2882 3003 


3125 


3246 


3368 


3489 


3611 


3732 


3854 


2 


6 


4097 4218 


4339 


4461 


4582 


4704 


4825 


4947 


5068 


2 


7 


5311 5432 


5554 


5675 


5796 


5918 


6039 


6161 


6282 


2 


8 


6525 6646 


6767 


6889 


7010 


7132 


7253 


7374 


7496 


2 


9 


7738 7860 


7981 


8102 


8224 


8345 


8466 


8588 


8709 


2 


3580 


8952 9073 


9194 


9315 


9437 


9558 


9679 


9801 


9922 


2 


1 


5540164 0286 


5540407 


0528 


5540650 


0771 


5540892 


1013 


5541135 


2 


2 


1377 1498 


1620 


1741 


1862 


1983 


2104 


2226 


2347 


2 


3 


2589 2710 


2832 


2953 


3074 


3195 


3316 


3438 


3559 


2 


4 


3801 3922 


4044 


4165 


4286 


4407 


4528 


4649 


4770 


2 


! 5 


5013 5134 


5255 


5376 


5497 


5618 


5740 


5861 


5982 


2 


1 6 


6224 6345 


6466 


6587 


6708 


6829 


6951 


7072 


7193 


121 


7 


7435 7556 


7677 


7798 


7919 


8040 


8161 


8282 


8403 




8 


8645 8766 


8887 


9008 


9130 


9251 


9372 


9493 


9614 




9 


9856 9977 


5550098 


0219 


5550340 


0461 


5550582 


0703 


5550824 




3590 


5551065 1186 


1307 


1428 


1549 


1670 


1791 


1912 


2033 




1 


2275 2396 


2517 


2638 


2759 


2880 


3001 


3121 


3242 




9 


3484 3605 


3726 


3847 


3968 


4089 


4210 


4330 


4451 




3 


4693 4814 


4935 


5056 


5176 


5297 


5418 


5539 


5660 




4 


5902 6022 


6143 


6264 


6385 


6506 


6627 


6747 


6868 




5 


7110 7231 


7351 


7472 


7593 


7714 


7835 


7955 


8076 




6 


8318 8438 


8559 


8680 


8801 


8921 


9042 


9163 


9284 




7 


9525 9646 


9767 


9887 


5560008 


0129 


5560249 


0370 


5560491 




8 


5560732 0853 


5560974 


1094 


1215 


1336 


1456 


1577 


1698 




9 


1939 2060 


2180 


2301 


2422 


2542 


2663 


2784 


2904 






1 2 


3 


4b 


5 


6 


T 


8 


9 


-__ 



LOGARITHMS 



SINES, COSINES, TANGENTS, AND COTANGENTS. 



64 






LOG. SINE 0°. 






[Table IJ.\\ 


// 


0' 


V 


2' 


3' 


4' 


5' 


6' 


7' 


'/ 





CO 


6'4637261 


6-7647561 


6-9408473 


7-0657860 


7-1626960 


7-2418771 


7-3088239 


60 


1 


4-6855749 


709047 


683602 


432534 


675918 


41412 


30818 


98567 


59; 


2 


9866049 


779665 


719347 


456462 


693901 


55817 


42832 


7-3108870 


58 


3 


5-1626961 


849154 


754800 


480259 


711810 


70173 


54813 


19149 


57 


4 


2876349 


917548 


789965 


503926 


729646 


84483 


66760 


29404 


56 


5 


3845449 


984882 


824849 


527465 


747408 


98745 


78675 


39635 


55 


6 


4637261 


6-5051188 


859454 


550878 


765099 


7-1712961 


90557 


49842 


54 


7 


5306729 


116497 


893786 


574164 


782717 


27131 


7-2502407 


60024 


53 


6 


5886649 


180838 


927848 


597327 


800264 


41254 


14225 


70183 


52 


9 


6398174 


244239 


961645 


620366 


817741 


55332 


26010 


80318 


51 


10 


6855749 


306729 


995182 


643284 


835148 


69364 


37764 


90430 


50 


11 


7269676 


368332 


6-8028461 


666082 


852485 


83351 


49485 


7-3200518 


49 


12 


7647561 


429074 


061488 


688760 


869753 


97293 


61176 


10583 


48 


13 


7995182 


488977 


094265 


711321 


886953 


7-1811190 


72835 


20624 


47 


14 


8317029 


548066 


126796 


733765 


904085 


25043 


84462 


30643 


46 


15 


8616661 


606361 


159086 


756094 


921149 


38853 


96059 


40638 


45 


16 


8896948 


663884 


191137 


778309 


938147 


52618 


7-2607625 


50610 


44 


17 


9160238 


720656 


222954 


800410 


955079 


66340 


19160 


60560 


43 


18 


9408474 


776695 


254539 


822400 


971945 


80018 


30664 


70487 


42 


19 


9643285 


832019 


285896 


844279 


988745 


93654 


42138 


80391 


41 


20 


9866049 


886648 


317029 


866048 


7-1005481 


7-1907247 


53582 


90272 


40! 


21 


6-0077942 


940599 


347939 


887709 


22153 


20797 


64996 


7-3300131 


39! 


22 


0279975 


993887 


378632 


909262 


38760 


34306 


763G0 


09968 


381 


23 


0473027 


6-6046529 


409109 


930708 


55305 


47772 


87734 


19783 


37 


24 


0657861 


098541 


439373 


952050 


71787 


61197 


99058 


29575 


35 


25 


0835149 


149938 


469428 


973287 


88206 


74580 


7-2710353 


39345 


26 


1005482 


200733 


499277 


994420 


7-1104564 


87923 


21619 


49094 


34 


27 


1169386 


250941 


528922 


7-0015451 


20860 


7-2001224 


32856 


58821 


33 


28 


1327329 


300575 


558365 


036381 


37095 


14485 


44063 


68525 


32 


29 


1479729 


349649 


587611 


057211 


53270 


27706 


55242 


78209 


31 


30 


1626961 


398174 


616661 


077941 


69385 


40886 


66392 


87870 


30' 


31 


1769366 


446162 


645518 


098572 


85440 


54027 


77514 


97511 


99' 


32 


1907248 


493627 


674184 


119107 


7-1201436 


67128 


88607 


7-3407130 


28i 


33 


2040888 


540578 


702663 


139544 


17374 


80189 


99672 


167^7 


27; 


34 


2170538 


587027 


730955 


159886 


33253 


93211 


7-2810708 


26304 


26i 


35 


2296429 


632985 


759065 


180132 


49074 


7-2106195 


21717 


35859 


25 


36 


2418774 


678461 


786994 


200285 


64838 


19140 


32698 


45394 


24 


37 


2537766 


723466 


814745 


220345 


80545 


32046 


43651 


54907 


23 


38 


2653585 


768009 


842319 


240313 


96195 


44914 


54577 


64400 


22 


39 


2766395 


812100 


869719 


260189 


7-1311789 


57744 


65475 


73872 


21 


40 


2876349 


855748 


896948 


279975 


27328 


70536 


76346 


83323 


20 


41 


2983587 


898962 


924007 


299671 


42811 


83290 


87190 


92754 


19 


42 


3088242 


941750 


950898 


319278 


58238 


96008 


98006 


7-3502165 


18 


43 


3190433 


984121 


977624 


338796 


73612 


7-2208688 


7-2908796 


11555 


17 


44 


3290275 


6-7026082 


6-9004187 


358228 


88931 


21331 


19560 


20925 


16: 


45 


3387874 


067641 


030588 


377573 


7-1404196 


33938 


30296 


30275 


15; 


46 


3483327 


108807 


056829 


396832 


19408 


46508 


41006 


39604 


14 


47 


3576727 


149586 


082913 


416006 


34566 


59041 


51690 


48914 


13! 


48 


3668161 


189986 


108841 


435096 


49672 


71539 


62347 


58203 


12 


49 


3757709 


230013 


134615 


454103 


64726 


84001 


72979 


67473 


11 


50 


3845449 


269675 


160237 


473026 


79727 


96427 


83584 


76723 


10 


51 


3931450 


308978 


185709 


491868 


94677 


7-2308818 


94164 


85954 


9 


52 


4015782 


347929 


211033 


510628 


7-1509576 


21173 


7-3004718 


95165 


8 


53 


4098507 


386533 


236209 


529307 


24423 


33494 


15246 


7-3604356 


7 


54 


4179686 


424797 


261241 


547906 


39221 


45779 


25749 


13528 


6 


55 


4259376 


462727 


286129 


566426 


53967 


58030 


36227 


22681 


5 


56 


4337629 


500328 


310375 


584868 


68664 


70246 


46679 


31814 


4 


57 


4414497 


537607 


335481 


603231 


83312 


82429 


57106 


40929 


3 


58 


4490029 


574569 


359948 


621517 


97910 


94577 


67509 


50024 


2: 


^9 


4564269 


611218 


384278 


639727 


7-1612459 


7-2406691 


77886 


59100 


1 


60 


4637261 


647561 


408473 


657860 


26960 


18771 


88239 


68157 





1" 


59' 


58' 


57' 


56' 


55' 1 


54' 


53' 


52' "II 




———- 




LOG. COSINE 89° 




ssssasi 


1 



Table ii.] 






LOG. TAN. 0°. 




65 II 


1" 0' 


I 1' 


2' 


3' 


4' 


5' 


6' 


7' 


//! 


—00 


6-4637261 


6-7647562 


6-9408475 


7-0657863 


7-1626964 


7-2418778 


7-3088248 


60 


1 4-6855749 


709047 


683603 


432536 


75921 


41417 


30825 


98576 


59 


2 9866049 


779666 


719347 


456464 


93904 


55821 


42839 


7-3108879 


58 


3 5- 1626961 


849154 


75480C 


480261 


7-0711813 


70178 


54819 


19158 


57 


4 2376349 


917549 


789966 


503928 


29649 


84488 


66767 


29413 


56 


5 3845449 


984882 


824849 


527467 


47412 


98750 


78682 


39644 


55 


6 4637261 


6-5051186 


859455 


550879 


65102 


7-1712966 


90564 


49851 


54 


7 5306729 


116497 


893786 


574166 


82720 


27136 


7-2502414 


60034 


53 


8 5886649 


180836 


927849 


597328 


7-0800268 


41259 


14231 


70193 


52 


9 6398174 


244240 


961646 


620368 


17744 


55337 


26017 


80328 


51 


10 6855749 


306729 


995183 


643286 


35151 


69369 


37771 


90440 


50 


11 7269676 


368332 


6-8028462 


666084 


52488 


83356 


49492 


7-3200528 


49 


12 7647561 


429074 


061489 


688762 


69756 


97298 


61183 


10592 


48 


13 7995182 


488977 


094266 


711323 


86956 


71811195 


72842 


20634 


47 


14 8317029 


548066 


126797 


733767 


7-0904088 


25049 


84469 


30652 


46 


115 8616661 


606361 


159087 


756096 


21153 


38858 


96066 


40648 


45 


16 8896948 


663885 


191138 


778311 


38151 


52623 


72607632 


50620 


44 


17 9160238 


720656 


222955 


800412 


55082 


66345 


19167 


60570 


43 


18 9408474 


776695 


254540 


822402 


71948 


80023 


30672 


70496 


42 


19 9643285 


832020 


285897 


844281 


88749 


93659 


42146 


80400 


41 


20 9866049 


886649 


317030 


866050 


7-1005484 


7-1907252 


53590 


90282 


40 


21 6-0077942 


940599 


347940 


887711 


22156 


20802 


65003 


7-3300141 


39 


22 0279975 


993887 


378633 


909264 


38764 


34311 


76387 


09978 


38 


23 0473027 


66046530 


409110 


930710 


55309 


47777 


87741 


19793 


37 


24 0657861 


098542 


439374 


952052 


71790 


61202 


99066 


29585 


36 


25 0835149 


149938 


469429 


973289 


88210 


74586 


7-2710361 


39356 


35 


26 1005482 


200733 


499278 


994422 


7-] 104567 


87928 


21627 


49104 


34 


27 1169386 


250941 


528923 


70015454 


20864 


7-2001230 


32863 


58831 


33 


28 1327329 


300576 


558367 


036383 


37099 


14491 


44071 


68536 


32 


•29 1479729 


349649 


587612 


057213 


53274 


27711 


55250 


78219 


3i; 


30 1626961 


398174 


616662 


077943 


69389 


40892 


66400 


87881 


30 


31 1769366 


446163 


645519 


098575 


85444 


54032 


77521 


97521 


29 


32 1907248 


493627 


674185 


119109 


7-1201440 


67133 


88615 


7-3407140 


28 


33 2040888 


540578 


702664 


139546 


17378 


80195 


99679 


16738 


27 


34 2170538 


587027 


730957 


159888 


33257 


93217 


7-2810716 


26314 


26 


35 2296429 


632985 


759066 


180135 


49078 


7-2106201 


21725 


35870 


25 


36 2418774 


678461 


786995 


200288 


64842 


19145 


32706 


45404 


24 


37 2537766 


723466 


814746 


220348 


80549 


32052 


43659 


54918 


23 


38 2653585 


768010 


842320 


240315 


96199 


44920 


54585 


64411 


22 


39 2766395 


812101 


869721 


260191 


7-1311793 


57750 


65483 


73883 


21 


40 2876349 


855749 


896949 


279977 


27332 


70542 


76354 


83334 


20 


41 2983587 


898963 


924008 


299673 


42815 


83296 


87198 


92765 


19 


42 3088242 


941751 


950900 


319280 


58242 


96014 


98015 


7-3502176 


18 


43 3190433 


984121 


977626 


338799 


73616 


7-2208694 


7-2908805 


11566 


17 


44 3290275 


6-7026082 


6-9004188 


358231 


88935 


21337 


19568 


20936 


16 


45 3387874 


067642 


030589 


377576 


7-1404200 


33944 


30304 


30286 


15 


46 3483327 


108808 


056830 


396835 


19412 


46514 


41015 


39615 


14 


47 3576727 


149587 


082914 


416009 


34570 


59048 


51698 


48925 


13 


48 3668161 


189987 


108842 


435099 


49676 


71545 


62356 


58215 


12 


49 3757709 


230014 


134617 


454105 


64730 


84007 


72987 


67485 


11 


50 3845449 


269676 


160239 


473029 


79732 


96433 


83593 


76735 


10 


51 3931450 


308979 


185711 


491870 


94681 


7-2308824 


94173 


85965 


9 


52 4015782 


347929 


211034 


510630 


7-1509580 


21180 


7-3004727 


95176 


8 


53 4098507 


386534 


236211 


529310 


24428 


33500 


15255 


7-3604368 


7 


54 4179686 


424798 


261242 


547909 


39225 


45786 


25758 


13540 


6 


55 4259376 


462728 


286130 


566429 


53972 


58036 


36235 


22692 


5 


56 4337629 


500329 


310876 


584871 


68669 


70253 


46688 


31826 


4 


57 4414497 


537608 


335482 


603234 


83316 


82435 


57115 


40940 


3 


58 4490029 


574570 


359950 


621520 


97914 


94583 


67517 


50035 


2 


59 4564269 


611219 


384280 


639730 


r-161-2464 


r -2406698 


77895 


59112 


1 


60 4637261 


647562 


408475 


657863 26964 


18778 


88248 


68169 





" 59' 


58' 


57' 


56' 55' 


54' 


53' 


53' 


" 








LOG. COTAN. 89°. 




^1 



6* 



1 66 






LOG. SINE 0°. 






Table n. || 


~ 


8' 


9' 


10' 


11' 


12' 


13' 


14' 


15' 


// 





7-3668157 


7-4179681 


7-4637255 


7-5051181 


7-5429065 


7-5776684 


7-6098530 


7-6398160 


60 


1 


77195 


87716 


44487 


57756 


35092 


82249 


7-6103697 


7-6402983 


59 


2 


86215 


95737 


51707 


64321 


41112 


87806 


08858 


07800 


58 


3 


95216 


7-4203742 


58916 


70876 


47123 


93356 


14012 


12612 


57 


4 


7-3704198 


11733 


66112 


77422 


53125 


98899 


19161 


17419 


56 


5 


13162 


19709 


73296 


83958 


59120 


7-5804435 


24304 


22221 


55 


1 6 


22107 


27670 


80469 


90483 


65106 


09964 


29440 


27017 


54 


7 


31034 


35617 


87629 


96999 


71084 


15486 


34571 


31808 


53 


8 


39943 


43549 


94778 


7-5103506 


77053 


21000 


39695 


36593 


52 


9 


48632 


51467 


7-4701915 


10002 


83015 


26508 


44813 


41373 


51 


10 


57705 


59370 


09041 


16489 


88968 


32009 


49926 


46149 


50 


11 


66559 


67259 


16154 


22966 


94913 


37503 


55032 


50918 


49 


12 


75396 


75134 


23257 


29434 


7-5500850 


4-2990 


60132 


55683 


48 


13 


84214 


82995 


30347 


35892 


06779 


48470 


65227 


60442 


47 


14 


93014 


90841 


37426 


42340 


12700 


53943 


70315 


65196 


46 


15 


7-3801796 


98673 


44493 


48779 


18613 


59409 


75397 


69945 


45 


16 


10561 


7-4306491 


51549 


55208 


24518 


64869 


80474 


74689 


44 


17 


19308 


14295 


58594 


61628 


30414 


70321 


85544 


79428 


43! 


18 


28038 


22085 


65627 


68038 


36303 


75767 


90609 


84161 


421 


19 


36750 


29861 


72649 


74439 


42184 


81206 


95668 


88889 


41 


20 


45444 


37624 


79659 


80830 


48057 


86638 


7-6200721 


93613 


40 


21 


54122 


45372 


86658 


87212 


53921 


92063 


05768 


98331 


39 


22 


62782 


53106 


93646 


93585 


59778 


97481 


10809 


7-6303043 


38 


23 


71424 


60827 


7-4800623 


99948 


65627 


7-5902893 


15844 


07751 


37 


24 


80050 


68534 


07588 


7-5206302 


71469 


08298 


20873 


12454 


36 


25 


88658 


76228 


14542 


12646 


77302 


13696 


25897 


17151 


35 


26 


97249 


83908 


21485 


18982 


83127 


19088 


30915 


21844 


34 


27 


7-3905824 


91574 


28417 


25308 


88945 


24473 


35927 


26531 


33 


28 


14381 


99227 


35338 


31625 


94755 


29851 


40933 


31214 


32 


29 


22922 


7-4406866 


42248 


37933 


7-5600557 


35223 


45934 


35891 


31 


30 


31446 


14492 


49147 


44231 


06352 


40588 


50928 


40563 


30 


31 


39953 


22104 


56035 


50521 


12138 


45946 


55917 


45231 


29 


32 


48444 


29703 


62913 


56801 


17917 


51298 


60901 


49893 


28 


33 


56918 


37289 


69779 


63073 


23689 


56643 


65878 


54550 


27 


34 


65375 


44862 


76634 


69335 


29452 


61981 


70850 


59203 


26 


l35 


73816 


52421 


83479 


75588 


35208 


67313 


75816 


63850 


25 


36 


S2241 


59968 


90313 


8L833 


40957 


72639 


60777 


68492 


24 


37 


90650 


67501 


97136 


88068 


46698 


77958 


85732 


73130 


23 


38 


99042 


75021 


7-4903949 


94295 


52431 


83270 


90681 


77762 


22 


39 


7-4007418 


82529 


10750 


7-5300512 


58157 


88576 


95624 


82390 


21 


40 


15778 


90023 


17541 


06721 


63875 


93876 


7-6300562 


87012 


20 


'41 


24121 


97504 


24322 


12920 


69585 


99169 


05495 


91630 


19 


142 


32449 


7-4504973 


31092 


19111 


75289 


7-6-004455 


10421 


96243 


18 


43 


40761 


12428 


37851 


25294 


80984 


09735 


15342 


7-6600850 


17 


44 


49057 


19871 


44600 


31467 


86672 


15009 


20258 


05453 


16 


45 


57337 


27302 


51339 


37631 


92353 


20277 


25168 


10052 


15 


46 


65601 


34719 


58067 


43787 


98026 


25538 


30073 


14645 


14 


47 


73850 


42124 


64784 


49934 


7-5703692 


30792 


34971 


19233 


13 


48 


82083 


49516 


71492 


56073 


09351 


36040 


39865 


23817 


12 


49 


90301 


56896 


78188 


62202 


15002 


41-282 


44753 


28395 


11 


50 


98503 


64263 


84875 


68324 


20646 


46518 


49635 


32969 


10 


51 


7-4106689 


71618 


91551 


74436 


26282 


51747 


54512 


37538 


9 


52 


14860 


78960 


98217 


80540 


31912 


56970 


59384 


42103 


8 


53 


23016 


86290 


r-5004873 


86635 


37533 


62187 


64250 


46662 


7 


54 


31156 


93607 


11519 


92722 


43148 


67397 


69110 


51217 


6 


55 


39282 


7-4600912 


18154 


98800 


48755 


72602 


73965 


55767 


5 


56 


47392 


08205 


24780 


7-5404870 


54356 


77800 


78815 


60312 


4 


57 


55487 


15486 


31395 


10931 


59949 


82991 


83659 


64852 


3 


58 


63567 


22754 


38000 


16984 


65534 


88177 


88498 


69388 


2 


59 


71631 


30011 


44595 


23029 


71113 


93356 


93332 


73919 


1 


60 


79681 


37255 


51181 


29065 


76684 


98530 


98160 


78445 





// 


5r 


50' 


49' 


48' 


47' 


46' 


45' 


44" 


// 




-. 







LOG. CO 


SINE 89= 


>^ 




1| 



Table II.] 






LOG. TAN. 0°. 






"^ll 


" 


8' 


9' 


10' 


11' 


12' 


13' 


14' 


15' 


" 





r-3668169 


7-4179696 


74637273 


7-5051203 


7-5429091 


7-5776715 


7-6098566 


7-6398201 


60 


1 


77207 


87731 


44506 


57778 


35119 


82280 


7-6103733 


7-6403024 


59 


2 


86227 


95752 


51726 


64343 


41138 


87837 


08894 


07842 


53 


3 


95228 


7-4203757 


58934 


70899 


47149 


93387 


14049 


12654 


57 


4 


7-3704210 


11748 


66130 


77444 


53152 


98930 


19197 


17461 


56 


5 


13174 


19724 


73315 


63980 


59147 


7-5804466 


24340 


22262 


55 1 


6 


22119 


27685 


80487 


90506 


65133 


09995 


29477 


27059 


54 


7 


31046 


35632 


87648 


97022 


71111 


15517 


34607 


31850 


53 


8 


39955 


43564 


94797 


7-5103528 


77080 


21032 


39732 


36635 


52 


9 


48845 


51482 


7-4701934 


10025 


83042 


26540 


44850 


41416 


51 


10 


57718 


59386 


09060 


16512 


88995 


32041 


49963 


46191 


50 


11 


66572 


67275 


16173 


22989 


94941 


37535 


55069 


50961 


49 


12 


75408 


75150 


23276 


29457 


7-5E00878 


43022 


60169 


55725 


48 


13 


84226 


83010 


30366 


35915 


06807 


48502 


65264 


60485 


47 


14 


93026 


90857 


37445 


42363 


12728 


53975 


70352 


65239 


46 


15 


7-3801809 


98689 


44513 


48802 


18640 


59441 


75435 


69988 


45 


16 


10574 


7-4306507 


51569 


55231 


24545 


64901 


80511 


74732 


44 


17 


19321 


14311 


58613 


61651 


30442 


70353 


85582 


79471 


43 


18 


28051 


22101 


65646 


68061 


36331 


75799 


90647 


84204 


42 


19 


36763 


29877 


72668 


74462 


42212 


81238 


95705 


88933 


41 


20 


45457 


37640 


79679 


80854 


48084 


86670 


7-6200758 


93656 


40 


21 


54134 


45388 


86678 


87236 


53949 


92096 


05805 


98374 


39 


22 


62794 


53123 


93666 


93608 


59806 


97514 


10847 


7-6503087 


38 


23 


71437 


60843 


7-4800642 


99972 


65656 


7-5902926 


15882 


07795 


37 


24 


80063 


68551 


07608 


7-5206326 


71497 


08331 


20911 


12497 


36 


25 


88671 


76244 


14562 


12670 


77330 


13730 


25935 


17195 


35 


26 


97263 


83924 


21505 


19006 


83156 


19121 


30953 


21888 


34 


27 


7 3905837 


91590 


28437 


25332 


88974 


24506 


35965 


26575 


33 


28 


14395 


99243 


35359 


31649 


94784 


29884 


40972 


31258 


32 


29 


22935 


7-4406882 


42269 


37957 


7-5600586 


35256 


45972 


35935 


31 


30 


31459 


14508 


49168 


44256 


06380 


40621 


50967 


40608 


30 


31 


39967 


22121 


56056 


50545 


12167 


45980 


55956 


45275 


29 


32 


48457 


29720 


62933 


56826 


17946 


51331 


60939 


49937 


28 


33 


56931 


37306 


69799 


63097 


23718 


56677 


65917 


54595 


27 


34 


65389 


44879 


76655 


69360 


29481 


62015 


70889 


59247 


26 


35 


73830 


52438 


83500 


75613 


35238 


67347 


75855 


63895 


25 


36 


82255 


59985 


90334 


81858 


40986 


72673 


80816 


68537 


24 


37 


90663 


67518 


97157 


88093 


46727 


77992 


85771 


73174 


23 


38 


99055 


75038 


7-4903969 


94319 


52460 


83304 


90720 


77807 


22 


39 


7-4007431 


82546 


10771 


7-5300537 


58186 


88611 


95664 


82435 


21 


40 


15791 


90040 


17562 


06745 


63904 


93910 


7-6300602 


87057 


20 


41 


24135 


97521 


24343 


12946 


69615 


99203 


05534 


91675 


19 


42 


32463 


7-4504990 


31113 


19137 


75318 


7-6004490 


10461 


96288 


18 


43 


40775 


12446 


37872 


25319 


81014 


09770 


15382 


7-6600896 


17 


44 


49071 


19889 


44621 


31492 


86702 


15044 


20298 


05499 


16 


45 


57351 


27319 


51360 


37657 


92383 


20311 


25208 


10097 


15 


46 


65616 


34737 


58088 


43813 


98056 


25572 


30113 


14690 


14 


47 


73864 


42141 


64806 


49960 


7-5703722 


30827 


35012 


19279 


13 


48 


82097 


49534 


71513 


56098 


09381 


36075 


39905 


23863 


12 


49 


90315 


56913 


78210 


62228 


15032 


41317 


44793 


28441 


11 


50 


98517 


64281 


84897 


68349 


20676 


46553 


49676 


33015 


10 


51 


7-4106703 


71635 


91573 


74462 


26313 


51782 


54553 


37585 


9 


52 


14875 


78978 


98239 


80566 


31942 


57005 


59424 


42149 


8 


53 


23030 


86308 


7-5004895 


86661 


37564 


62222 


64290 


46709 


7 


54 


31171 


93625 


11541 


92748 


43179 


67433 


69151 


51263 


6 


55 


39296 


7-4600930 


18176 


98826 


48786 


72637 


74006 


65813 


5 


56 


47406 


08223 


24802 


7-5404896 


54386 


77835 


78856 


60359 


4 


57 


55501 


15504 


31417 


10958 


59979 


83027 


83700 


64899 


3 


58 


63581 


22773 


38022 


17011 


65565 


88213 


88539 


69435 


2 


59 


71646 


30030 


44618 


23055 


71144 


93392 


93373 


73966 


1 


60 


79696 


37273 


51203 


29091 


76715 


98566 


98201 


78492 





// 


5r 


W 


49' 


48' 


47' 


46' 


45' 


44' 


'-' 










LOG. CO 


TAN. 89^ 


• 




-^ — ^1 



i68 




- 


LOG. 


SINE 0=. 






[Table -n.\ 


" 


16- 


17' 


IS' 


19' 


20' 


21' 


f 22' 


23' "' 


i 


7-667S445 


7-6941733 


7-7139966 


7-7424775 


7-7&47537 


7-7S59427 


7-806145^. 


7-S254507 60' 


1 


82967 


45933 


93936 


28533 


51154 


62372 


64747 


57653 59 


1 2 


87434 


50240 


93001 


32333 


54769 


66315 


68033 


60797i53 


3 


91996 


54437 


7-7202013 


36139 


53330 


69755 


71317 


63933:57 


4 


96503 


58730 


06021 


39937 


61939 


73192 


74599 


67077156 


'; 5 


7-6701006 


62969 


10026 


43731 


65594 


76627 


77873 


70214:55 


6 


05504 


67204 


14027 


47573 


69197 


80053 


81154 


73343i54 


! 7 


09998 


71435 


13024 


51360 


72797 


83483 


84428 


76431 153 


!8 


14486 


75662 


22017 


55145 


76393 


86914 


87699 


79611 


52 


I 9 


13970 


79334 


26007 


53926 


79987 


90337 


90968 


82738 


51| 


jio 


23450 


84103 


29993 


62705 


83577 


93758 


94235 


85364 


50i 


ill 


27925 


88317 


33976 


66479 


67165 


97177 


97499 


639371491 


12 


32395 


92523 


37955 


70251 


90750 


7-7900592 


7-S100761 


92103 


43 


13 


36361 


96734 


41930 


74019 


94332 


04005 


04020 


95227 


47; 


14 


41322 


7-7000936 


45902 


77734 


97910 


07415 


07227 


93343 


46. 


15 


45779 


05134 


49369 


81546 


7-7701436 


10323 


10531 


7-8301453 


45, 


116 


50231 


09323 


53334 


65304 


05059 


14228 


13783 


04570 


44i 


il7 


54673 


13513 


57794 


89059 


08629 


17630 


17032 


07630 


43i 


il8 


59121 


17704 


61752 


92311 


12196 


21029 


20279 


10737 


421 


;19 


63559 


21336 


65705 


96560 


15760 


24426 


23524 


13393 


4l| 


i20 


67993 


26064 


69655 


7-7500306 


19322 


27820 


26766 


16996 


40' 


21 


72422 


30233 


73601 


04043 


22330 


31212 


30006 


20097 


39 


122 


76847 


34407 


77544 


07737 


26435 


34601 


33243 


23195 


33 


i23 


81267 


38573 


81433 


11523 


29983 


37987 


36478 


26292 


37,: 


124 


85633 


42735 


85419 


15255 


33537 


41371 


39711 


29336 


36: 


125 


90094 


46393 


89351 


13935 


37034 


44752 


42941 


32473 


35 


^26 


94501 


51047 


93279 


22711 


40623 


43130 


46168 


35563 


34' 


^27 


98904 


55197 


97204 


26434 


44169 


51506 


49394 


33656 


33 


,23 


7-6303302 


59343 


7-7301125 


30154 


47707 


54879 


52617 


41741 


32 


29 


07695 


63435 


05043 


33371 


51242 


53250 


55837 


44825 


311 


30 


12084 


67623 


03957 


37534 


54774 


61617 


59055 


47906 30'| 


31 


16469 


71757 


12363 


41294 


58303 


64933 


62271 


50935 


29 


132 


20849 


75837 


16776 


45001 


61830 


63345 


65484 


54062 




,33 


25224 


80014 


20679 


48705 


65354 


71705 


68695 


57136 


27i 


34 


29596 


84136 


24579 


52406 


68374 


75063 


71904 


60209 


26' 


35 


33963 


88254 


28476 


56104 


72392 


78418 


75110 


63279 


25 


36 


38325 


92369 


32369 


59793 


75907 


81770 


78314 


66347 


24 


37 


42693 


96430 


36259 


63490 


79420 


65120 


81516 


69413 


23 


38 


47037 


7-7100536 


40145 


67178 


62929 


83467 


84715 


72477 


22 


39 


51337 


04639 


44023 


70363 


86436 


91311 


87912 


75533 


21 


40 


55732 


08783 


47903 


74545 


69939 


95153 


91106 


78598 


20 


41 


60072 


12833 


51733 


73224 


93440 


93493 


94298 


81655 


19 


|42 


64409 


16975 


55656 


81900 


96933 


7-SO01330 


97433 


84710 


18 


43 


68741 


21062 


59525 


65572 


7-7500434 


05164 


7'S200676 


87763 


17 


{44 


73069 


25146 


63390 


89242 


03926 


03496 


03361 


90814 


16 


45 


77392 


29225 


67252 


92905 


07416 


11325 


07043 


93863 


15 


'46 


81711 


33301 


71111 


96572 


10903 


15151 


10224 


96909 


14 


:47 


86026 


37373 


74966 


7-7600232 


14387 


18475 


13402 


99954 


13 


148 


90337 


41442 


73313 


03339 


17863 


21797 


16573 


7-S402996 


12 


;49 


94643 


45506 


62666 


07543 


21347 


25116 


19751 


06036 


11 


'so 


93945 


49567 


86511 


11194 


24322 


23432 


22922 


09074 


10 


151 


7-6903243 


53624 


90353 


14342 


23295 


31746 


26091 


12110 


9 


,52 


07536 


57677 


94191 


13437 


31765 


35058 


29253 


15144 


8 


15.3 


11S26 


61726 


93026 


22129 


35233 


38367 


32422 


18176 


7 


'54 


16111 


65772 


7-7401357 


25768 


38697 


41673 


35534 


21205 


6' 


!55 


20392 


69314 


05635 


29403 


42159 


44977 


38743 


24233 


5 


,56 


24663 


73352 


09510 


33036 


45613 


48278 


41901 


27253 


4 


,57 


23941 


77836 


13331 


36666 


49075 


51577 


45056 


30231 


3 


53 


33209 


81917 


17149 


40292 


52523 


54373 


43209 


33302 


2 


59 


37473 


85943 


20964 


43916 


55979 


58167 


51359 


36321 


1 


60 


41733 


89966 


24775 


47537 


59427 


61453 


54507 


39333 





''' 


43' 


42' 


41' 


40' 


39' 


38' 


37' 


36' 




L 








LOG, COS 


INE 89° 






1 



\ Table 11.] 






LOG. TAN. 0°. 






H 


r77 


16' 


17' 


18' 


19' 


20' 


21' 


22' 


23' 


// 





7-6678492 


7-6941786 


7-7190026 


7-7424841 


7-7647610 


7-7859508 


7-8061547 


7-8254604 


60 


1 


83014 


46042 


94045 


28649 


51228 


62954 


64836 


57750 


59 


2 


87531 


50293 


98061 


32454 


54843 


66396 


68123 


60894 


58 


3 


92043 


54541 


7-7202073 


36255 


58454 


69836 


71407 


64036 


57 


4 


96551 


58784 


06081 


40053 


62063 


73274 


74688 


67175 


56 


5 


7-6701053 


63023 


10086 


43848 


65669 


76708 


77967 


70312 


55 


6 


05552 


67258 


14087 


47640 


69271 


80140 


81244 


73446 


54 


7 


10045 


71489 


18084 


51428 


72871 


83569 


84518 


76579 


53 


8 


14534 


75716 


22078 


55212 


76468 


86996 


87789 


79709 


52 


9 


19018 


79938 


26068 


58994 


80061 


90420 


91059 


82837 


51 


10 


23498 


84157 


30054 


62772 


83652 


93841 


94325 


85962 


50 


11 


27973 


88371 


34037 


66547 


87240 


97259 


97590 


89086 


49 


12 


32443 


92582 


38016 


70319 


90825 


7-7900675 


7-8100851 


92207 


48 


13 


36909 


96788 


41991 


74087 


94407 


04088 


04111 


95326 


47 


14 


41371 


7-7000990 


45963 


77852 


97986 


07498 


, 07368 


98443 


46 


15 


45827 


05189 


49931 


81614 


7-7701562 


10906 


10622 


7-8301557 


45 


16 


50279 


09383 


53895 


85372 


05135 


14311 


13874 


04669 


44 


17 


54727 


13573 


57856 


8912S 


08705 


17713 


17124 


07779 


43 


18 


59170 


17759 


61813 


92880 


12272 


21113 


20371 


10887 


42 


19 


63608 


21941 


65767 


96629 


15836 


24510 


23615 


13992 


41 


20 


68042 


26119 


69717 


7-7500374 


19398 


27904 


26858 


17096 


40 


21 


72471 


30293 


73663 


04117 


22956 


31296 


30098 


20197 


39 


22 


76896 


34463 


77606 


07856 


26512 


34685 


33335 


23296 


38 


23 


81317 


38629 


81545 


11592 


30064 


38071 


36570 


26392 


37 


24 


85733 


42791 


85481 


15325 


33614 


41455 


39803 


29487 


36 


25 


90144 


46949 


89413 


19054 


37161 


44836 


43033 


32579 


35 


26 


94551 


51103 


93342 


22780 


40705 


48215 


46261 


35669 


34 


27 


98953 


55253 


97267 


26504 


44246 


51590 


49486 


38757 


33 


28 


7-6803351 


59399 


7-7301188 


30224 


47784 


54964 


52709 


41843 


32 


29 


07745 


63541 


05106 


33940 


51319 


58334 


55930 


44926 


31 


30 


12134 


67679 


09020 


37654 


54851 


61702 


59148 


48007 


30 


31 


16519 


71813 


12931 


41364 


58381 


65068 


62364 


51087 


29 


32 


20899 


75944 


16839 


45072 


61907 


68431 


65578 


54163 


28 


33 


25275 


80070 


20742 


48776 


65431 


71791 


68789 


57238 


27 


34 


29646 


84193 


24643 


52477 


68952 


75148 


71998 


60311 


26 


35 


34013 


88311 


28540 


56174 


72470 


78503 


75204 


63381 


25 


36 


38376 


92426 


32433 


59869 


75985 


81856 


78408 


66449 


24 


37 


42734 


96537 


36323 


63560 


79498 


85206 


81610 


69515 


23 


38 


47088 


7-7100643 


40209 


67249 


83007 


88553 


84809 


72580 


22 


39 


51438 


04746 


44092 


70934 


86514 


91898 


88006 


75641 


21 


40 


55783 


08846 


47972 


74616 


90018 


95240 


91201 


78701 


20 


41 


60124 


12941 


51848 


78295 


93519 


98579 


94393 


81758 


19 


42 


64460 


17032 


55720 


81971 


97017 


7-8001916 


97583 


84813 


18 


43 


68792 


21120 


59589 


85644 


7-7800513 


05251 


7-8200770 


87867 


17 


44 


73120 


25203 


63455 


89313 


04005 


08582 


03956 


90918 


16 


45 


77444 


29283 


67317 


92980 


07495 


11912 


07139 


93966 


15 


46 


81763 


33359 


71176 


96643 


10982 


15238 


10319 


97013 


14 


47 


86078 


37432 


75031 


7-7600304 


14466 


18563 


13497 


7-8400058 


13 


48 


90389 


41500 


78883 


03961 


17948 


21884 


16673 


03100 


12 


49 


94695 


45565 


82731 


07615 


21426 


25203 


19847 


06140 


11 


50 


98997 


49625 


86577 


11266 


24902 


28520 


23018 


09179 


10 


51 


7-6903295 


53682 


90418 


14915 


28375 


31834 


26187 


12215 


9 


52 


07589 


57736 


94257 


18560 


31845 


35146 


29354 


15249 


8 


53 


11878 


61785 


98091 


22202 


35313 


38455 


32518 


18280 


7 


54 


16163 


65831 


7-7401923 


25840 


38778 


41761 


35680 


21310 


6 


55 


20444 


69873 


05751 


29476 


42240 


45065 


38840 


24338 


5 


56 


24721 


73911 


09576 


33109 


45699 


48366 


41997 


27363 


4 


57 


28993 


77945 


13397 


36739 


49155 


51665 


45153 


30387 


3 


58 


33262 


81976 


17215 


40366 


52609 


54962 


48305 


33408 


2 


59 


37526 


86003 


21030 


43989 


56060 


58256 


51456 


36427 


1 


60 


41786 


90026 


24841 


47610 


59508 


61547 


54604 


39444 





// 


43' 


42' 


41' 


40' 


39' 


38' 


37' 


36' 


" 


.. 




_______ 




LOG. CO 


rAN. 89^ 


• 




-1 



70 






LOG. SINE 0°. 


^^'^'^^'"^ 




[Tabfe II. 




„ 24' 


( 25' 


1 26' 


27' 


28' 


29' 


30' 


1 31' 1 " 




7-8439338|7-8616623 7-878695: 


7-8950854 


7-9108793 


7-9261190 


7-940841C 


7-9550S19'60 




1 4235. 


5 19517J 89736 


53534 


11376 


63685 


10831 


53153J59 




2 4536( 


i 2241( 


) 92517 


56212 


13960 


66179 


13241 


554S658 




3 4837' 


r 2530C 


95297 


58889 


16542 


68671 


15651 


57818'57 




4 5138c 


2818C 


98075 


61564 


19121 


71162 


18059 


60149156 

62478155 




5 54395 


31075 


7-8500850 


64237 


21699 


73651 


20465 




6 57396 


3396C 


03625 


66909 


24276 


761'39 


22871 


64806'54 




7 60396 


36843 


06397 


69579 


26851 


78626 


25275 


67133 .'^S 




8 63399 


39723 


09167 


72248 


29425 


81111 


27677 


69458 


52 




9 66397 


42602 


11936 


74914 


31997 


83595 


30079 


71782 


51 




10 69393 


45479 


14703 


77580 


34567 


86077 


32479 


74105 


50 




11 72387 


48354 


17469 


80243 


37136 


88558 


34877 


76427 


49 




12 75379 


51228 


20232 


82905 


39704 


91037 


37275 


78747 


48 




13 78369 


54099 


22994 


85565 


42269 


93516 


39671 


81067 


47 




14 81357 


56968 


25754 


88224 


44834 


95992 


42066 


83385 


46 




15 84343 


59836 


28512 


90881 


47397 


98467 


44459 


85702 


45 




16 87326 


62702 


31269 


93536 


49958 


7-9300941 


46851 


88017 


44 




17 90308 


65565 


34023 


96190 


52518 


03414 


49242 


90331 


43 




18 93288 


68427 


36776 


98842 


55076 


05885 


51631 


92645 


42 




19 96265 


71287 


39528 


7-9001493 


57633 


08354 


54019 


94956 


41 




20 99241 


74145 


42277 


04141 


60189 


10823 


56406 


97267 


40 
39 




21 7-8502215 


77001 


45025 


06789 


62743 


13289 


58792 


99576 




22 05186 


79856 


47771 


09434 


65295 


15755 


61176 


7-9601885 


38 




23 08156 


82708 


50515 


12078 


67846 


18219 


63559 


04192 


37 




24 11123 


85559 


53258 


14721 


70395 


20682 


65940 


06497 


36 




25 14088 


86408 


55999 


17362 


72943 


23143 


68321 


08802 


35- 




26 17052 


91254 


58738 


20001 


75489 


25603 


70700 


11105 


34; 




27 20013 


94099 


61475 


22639 


78034 


28061 


73077 


13407 


33 




28 22973 


96942 


64211 


25275 


80578 


30518 


75454 


15708 


32 




29 25930 


99784 


66945 


27909 


83120 


32974 


77829 


18008 


31 




30 28885 


7-8702623 


69677 


30542 


85660 


35428 


80203 


20306 


30 




31 31839 


05461 


72407 


33173 


88199 


37881 


82575 


22603 


29 




32 34790 


08296 


75136 


35803 


90736 


40332 


84946 


24899 


28 




33 37739 


11130 


77863 


38431 


93272 


42783 


87316 


27194 


27 




34 40687 


13962 


80589 


41057 


95807 


45231 


89685 


29487 


26 




35 43632 


16792 


&3312 


43682 


98340 


47679 


92052 


31780 


25 




36 46575 


19621 


86034 


46305 


7-9200871 


50125 


94418 


34071 


24 




37 49517 


22447 


88754 


48927 


03401 


52569 


96783 


36361 


23i 




38 52456 


25272 


91473 


51547 


05930 


55012 


99146 


38649 


22, 




39 55393 


28095 


94190 


54166 


08457 


57-454 


7-9501508 


40937 


21 




40 58329 


30916 


96905 


56783 


10983 


59895 


03869 


43223 


20 




41 61262 


33735 


99618 


59398 


13507 


62334 


06229 


45508 


19 




42 64193 


36552 


7-8902330 


62012 


16030 


64772 


08587 


47792 


18 




43 67123 


39367 


05040 


64624 


18551 


67208 


10944 


50075 


17 




44 70050 


42181 


07749 


67235 


21071 


69643 


13300 


52356 


16 




45 72976 


44993 


10455 


69844 


23589 


72077 


15654 


54637 


15 




46 75899 


47803 


13160 


72451 


26106 


74509 


18008 


56916 


14i 




47 78821 


50611 


15864 


75057 


28621 


76940 


20360 


59194 


13 




48 81740 


53417 


18565 


77662 


31135 


79369 


22710 


61470 


12 




49 84658 


56222 


21265 


80265 


33648 


81798 


25060 


63746 


11 




50 87574 


59025 


23963 


82866 


36159 


84224 


27408 


66020 


10 




51 90487 


61826 


26660 


85466 


38668 


86650 


29755 


68293 


9 




52 93399 


64625 


29355 


88064 


41177 


89074 


32100 


70565 


8 




53 96309 


67422 


32048 


90660 


43683 


91497 


34444 


72836 


7 




54 99217 


70218 


34740 


93256 


46188 


93918 


36787 


75106 


6 




55 7-8602123 


73011 


37430 


95849 


48692 


96338 


39129 


77374 


5 




56 05027 


75803 


40118 


98441 


51195 


98757 


41470 


79641 


4 




157 07929 


78594 


42804 


7-9101031 


53696 


7-9401175 


43809 


81907 


3 




58 10829 


81382 


454S9 


03620 


56195 


03591 


46147 


84172 






59 13727 


84168 


48173 


06208 


58693 


06005 


46484 


86436 


1 




60 16623 


86953 


50854 


08793 


61190 


08419 


50819 


88698 






: '' 35' 


34' 


33' 


32' 


31' 


30' 


39' 


28' 


" 










LOG. CO 


SINE 89° 


• 


______ 


^1 





Table II.] 






LOG. TAN 0°. 71 




1 /" 


24' 


25' 


26' 


27' 


28' 


1 29' 


30' 


31' 


/ 







7'8439444 


7-8616738 


7-8787077 


7.8950988 


7-9108938 


7-9261344 


7-9408584 


7-9550996 


60 




1 


42459 


19632 


89861 


53668 


7-9111522 


3840 


7-9410996 


3330 


59 




2 


Ab4ri2 


22525 


92642 


56347 


4105 


6333 


3407 


5663 


58 




3 


48483 


25415 


95422 


59023 


6686 


8826 


5817 


7995 


57 




4 


51492 


28304 


98199 


61699 


9266 


7-9271317 


8225 


7-9560326 


56 




5 


54498 


31191 


7-8800975 


64372 


7-9121844 


3807 


7-9420632 


2655 


55 




6 


57503 


34076 


03750 


67044 


4421 


6295 


3037 


4984 


54 




7 


60505 


36958 


06522 


69714 


6996 


8782 


5441 


7310 


53 




8 


63506 


39839 


09293 


72383 


9570 


7-9281267 


7844 


9636 


52 




9 


66504 


42719 


12062 


75050 


7-9132142 


3751 


7-9430246 


7-9571961 


51 




10 


69500 


45596 


14829 


77715 


4713 


6233 


2646 


4284 


50 




11 


72494 


48471 


17594 


80379 


7282 


8714 


5045 


6606 


49 




12 


75487 


51344 


20358 


83041 


9850 


7-9291194 


7442 


8926 


48 




13 


78477 


54216 


23120 


85701 


7-9142416 


3672 


9839 


7-9581246 


47 




14 


81465 


57085 


25880 


88360 


4980 


6149 


7-9442233 


3564 


46 




15 


84451 


59953 


• 28639 


91017 


7543 


8625 


4627 


5881 


45 




16 


87435 


62819 


31395 


93673 


7-9150105 


7-9301099 


7019 


8197 


44 




17 


90416 


65683 


34150 


96327 


2665 


3571 


9410 


7-9590511 


43 




18 


93396 


68545 


36903 


98979 


5224 


6043 


7-9451800 


2825 


42 




19 


96374 


71405 


39655 


7-9001630 


7781 


8512 


4188 


5137 


41 




20 


99350 


74263 


42404 


04279 


7-9160336 


7-9310981 


6575 


7447 


40 




21 


7-8502323 


77120 


45152 


06926 


2890 


3448 


8961 


9757 


39 




22 


05295 


79974 


47899 


09572 


5443 


5913 


7-9461345 


7-9602065 


38 




23 


08265 


82827 


50643 


12216 


7994 


8378 


3728 


4373 


37 




24 


11232 


85677 


53386 


14859 


7-9170543 


7-9320840 


6110 


6678 


36 




25 


14198 


88526 


56127 


17500 


3091 


3302 


8491 


8983 


35 




26 


17161 


91373 


58866 


20139 


5638 


5762 


7-9470870 


7-9611287 


34 




27 


20123 


94218 


61604 


22777 


8183 


8220 


3248 


3589 


33 




28 


23083 


97062 


64339 


25413 


7-9180727 


7-9330678 


5624 


5890 


32 




29 


26040 


99903 


67074 


28048 


3269 


3133 


8000 


8190 


31 




30 


28996 


■■S702743 


69806 


30681 


5809 


5588 


7-9480374 


7-9620488 


30 




31 


31949 


05580 


72537 


33312 


8348 


8041 


2746 


2786 


29 




32 


34900 


08416 


75266 


35942 


7-9190886 


7-9340493 


5118 


5082 


28 




33 


37850 


11250 


77993 


38570 


3422 


2943 


7488 


7377 


27 




34 


40797 


14082 


80718 


41197 


5957 


5392 


9856 


9670 


26 




35 


43743 


16913 


83442 


43822 


8490 


7839 


7-9492224 


7-9631963 


25 




36 


46686 


19741 


86164 


46445 


7-9201022 


7-9350286 


4590 


4254 


24 




37 


49628 


22568 


88885 


49067 


3552 


2730 


6955 


6544 


23 




38 


52567 


25393 


91603 


51687 


6081 


5174 


9319 


8833 


22 




39 


55505 


28215 


94320 


54306 


8608 


7616 


7-9501681 


7-9641121 


21 




40 


58440 


31037 


97036 


56923 


7-9211134 


7-9360057 


4042 


3408 


20 




41 


61374 


33856 


99749 


59539 


3658 


2496 


6402 


5693 


19 




42 


64305 


36673 


7-8902461 


62153 


6181 


4934 


8760 


7977 


18 




43 


67235 


39489 


05171 


64765 


8702 


7370 


7-9511118 


7-9650260 


17 




44 


70163 


42303 


07880 


67376 


7-9221222 


9805 


3474 


2541 


16 




45 


73088 


45115 


10587 


69985 


3741 


7-9372239 


5828 


4822 


15 




46 


76012 


47925 


13292 


72593 


6258 


4672 


8182 


7101 


14 




47 


78934 


50733 


15995 


75199 


8774 


7103 


7-9520534 


9379 


13 




48 


81853 


53540 


18697 


77804 


7-9231288 


9533 


2885 


7-9661656 


12 




49 


84771 


56344 


21397 


80407 


3800 


7-9381961 


5234 


3932 


11 




50 


87687 


59147 


24096 


83008 


6312 


4388 


7582 


6206 


10 




51 


90601 


61949 


26792 


85608 


8821 


6814 


9929 


8480 


9 




52 


93513 


64748 


29487 


88207 


7-9241330 


9238 


7-05322V5 


7-9670752 


8 




53 


96423 


67545 


32181 


90803 


3836 


7-9391661 


4620 


3023 


7 




54 


99331 


70341 


34873 


93399 


6342 


4083 


6963 


5293 


6 




55 


7'8602237 


73135 


37563 


95992 


8846 


6503 


9305 


7561 


5 




56 


05141 


75927 


40251 


98584 


7-9251348 


8922 


7-9541646 


9829 


4 




57 


08043 


78717 


42938 


79101175 


3850 


7-9401339 


3985 


7-9682095 


3 




58 


10943 


81506 


45623 


03764 


6349 


3756 


6323 


4360 


2 




59 


13841 


84293 


48306 


06352 


8847 


6170 


8660 


6624 


I 




60 


16738 


87077 


50988 


08938 


7-9261344 


8584 7-95509961 88861 







' 


35' 


34' 


33' 


32' 


31' 


30' 


29' 1 


28' 1 


/ 





LOG. COTAN. 89^. 



72 LOG. SINE. [Ta6/e II. Il 


" 


32' 


33' 


34' 


35' 


36' , 37' 


38' 


39' 


" 





7-9688698 


7-9822334 


7-9951980 


8-0077867 


80200207 


8-0319195 


8-0435009 


80547814 


60 


1 


7%90960 


4527 


4108 


9934 


2217 


8-0321150 


6913 


9670 


59 


2 


3220 


6718 


6235 


8-0082001 


4226 


3105 


8816 


8-0551524 


58 


3 


5479 


8909 


8361 


4066 


6234 


5059 


8-0440719 


3378 


57 


4 


7736 


7-9831098 


7-9960487 


6131 


8242 


7012 


2621 


5231 


56 


5 


9993 


3287 


2611 


8194 


8-0210248 


8965 


4522 


7084 


55 


6 


7-9702248 


5474 


4734 


8-0090257 


2253 


8-0330916 


6422 


6935 


54 


7 


4503 


7660 


6856 


2318 


4258 


2866 


8321 


80560786 


53 


8 


6756 


9845 


8977 


4379 


6261 


4816 


80-450220 


2636 


52 


9 


9008 


7-9842029 


7-9971097 


6439 


8264 


6765 


2117 


4485 


51 


10 


7-9711258 


4212 


3216 


8497 


8-0220266 


8713 


4014 


6333 


50 


11 


3508 


6394 


5334 


8-0100555 


2267 


8-0340660 


5910 


8181 


49 


12 


5756 


8574 


7451 


2612 


4267 


2606 


7805 


8-0570028 


48 


13 


8004 


7-9850754 


9566 


4668 


6266 


4551 


9700 


1874 


47 


14 


7-9720250 


2933 


7-9981681 


6722 


8264 


6495 


8-0461593 


3719 


46 


15 


2495 


5110 


3795 


8776 


8-0230261 


8439 


3486 


5563 


451 


16 


4738 


7286 


5908 


8-0110829 


2257 


8-0350382 


5378 


7407 


44 


17 


6981 


9461 


8020 


2881 


4252 


2323 


7269 


9250 


43 


18 


9222 


7-9861636 


7-9990130 


4932 


6247 


4264 


9159 


8-0581092 


42 


19 


7-9731463 


3809 


2240 


6982 


8240 


6204 


8-0471048 


2933 


41 


20 


3702 


5981 


4349 


9031 


8-0240233 


8143 


2937 


4774 


40 


21 


5940 


8151 


6456 


8-0121079 


2224 


8-0360082 


4825 


6614 


39 


22 


8177 


7-9870321 


8563 


3126 


4215 


2019 


6712 


8453 


38 


23 


7-9740412 


2490 


8-0000669 


5172 


6205 


3956 


8598 


8-0590291 


37 


24 


2647 


4658 


2773 


7217 


8194 


5892 


8-0480483 


2128 


36 


25 


4880 


6824 


4877 


9261 


8-0250182 


7826 


2368 


3965 


35 


26 


7113 


8989 


6979 


8-0131304 


2169 


9760 


4251 


5801 


34 


27 


9344 


7-9881154 


9081 


3347 


4155 


8-0371693 


6134 


7636 


33 


28 


7-9751574 


3317 


8-00m81 


5388 


6140 


3626 


8016 


9470 


32 


29 


3802 


5479 


3281 


7428 


8125 


5557 


9897 


8-0601304 


31 


30 


6030 


7641 


5379 


9468 


8-0260108 


7488 


8-0491778 


3137 


30 


31 


8257 


9801 


7477 


8-0141506 


2091 


9417 


3657 


4969 


29 


32 


7-9760482 


7-9891960 


9573 


3543 


4072 


80381346 


5536 


6800 


28 


33 


2706 


4117 


80021669 


5580 


6053 


3274 


7414 


8630 


27 


34 


4929 


6274 


3763 


7615 


8033 


5201 


9291 


8-0610460 


26 


35 


7151 


8430 


5856 


9650 


80270012 


7128 


8-0501167 


2289 


25 


36 


9372 


7-9900585 


7949 


8-0151684 


1990 


9053 


3043 


4117 


24 


37 


7-9771592 


2738 


8-0030040 


3716 


3967 


8-0390978 


4918 


5944 


23 


38 


3810 


4891 


2131 


5748 


5943 


2901 


6792 


7771 


22 


39 


6028 


7043 


4220 


7779 


7919 


4824 


8665 


9597 


21 


40 


8244 


9193 


6308 


9808 


9893 


6746 


8-0510537 


8 0621422 


20 


41 


7-9780459 


7-9911342 


8396 


8-0161837 


8-0281867 


8667 


2408 


3246 


19 


42 


2673 


3491 


8-0040482 


3865 


3839 


8-0400588 


4279 


5070 


18 


43 


4886 


5638 


2568 


5892 


5811 


2507 


6149 


6892 


17 


44 


7098 


7784 


4652 


7918 


7782 


4426 


8018 


8714 


16 


45 


9309 


9929 


6735 


9943 


9752 


6343 


9886 


8-0630536 


15 


46 


7-9791518 


7-9922073 


8818 


8-0171967 


8-0291721 


8260 


8-0521754 


2356 


14 


47 


3726 


4216 


8-0050899 


3991 


3689 


8-0410176 


3620 


4176 


13 


48 


5934 


6358 


2979 


6013 


5656 


2092 


5486 


5995 


12 


49 


8140 


8499 


5059 


8034 


7623 


4006 


7351 


7813 


111 


50 


7-9800345 


7-9930639 


7137 


8-0180055 


9588 


5920 


9216 


9630 


10 1 


51 


2549 


2778 


9215 


2074 


8-0301553 


7832 


80531079 


8-0641447 


9 


52 


4752 


4915 


80061291 


4093 


3517 


9744 


2942 


3263 


8 


53 


6953 


7052 


3366 


6110 


5479 


8-0421655 


4803 


5078 


7 


54 


9154 


9188 


5441 


8127 


7441 


3565 


6665 


6893 


6 


55 


7-9811353 


7-9941322 


7514 


80190142 


9403 


5475 


8525 


8706 


5 


56 


3552 


3456 


9587 


2157 


80311363 


7383 


8-0540384 


8-0650519 


4 


57 


5749 


5588 


8-0071658 


4171 


3322 


9291 


2243 


2331 


3 


58 


7945 


7720 


3729 


6184 


5280 


8-0431198 


4101 


4143 


2 


59 


7-9820140 


9850 


5798 


8196 


7238 


3104 


5958 


5953 


i: 


60 


2334 


7-9951980 


7867 


30200207 


9195 


5009 


7814 


7763 0| 


" 


27' 


26' 


25' 


24' 


23' 


22' 


21' 


20' "1 


LOG. COSINE 89''. j| 



iTable ii.] log. tan. 0°. 73 




'' 32' 


33' 


34' 


35' [ 36' 


1 37' 1 38' 


39' 1" 




7-96SS88 


6 7-982253 


i 7-995219 


2 8-0078092 8-0200445 8-0319446 8-043527^ 


1 8^0548094 60 




1 7-969114 


3 472 


7 432 


3 8-0OS0159 2455 80321402 717^ 


) 9949,59 




2 340 


3 691 


9 6445 


3 2226 4465 3357 908^ 


J 8-0551804 58 




3 566 


7 911 


T 857^ 


I 4292 6473 531] 


8-044098E 


3658 57 




4 792 


5 7-983129 


3 7-996070( 


) 6357 848 


726£ 


) 2887 


5512 56 




5 7-970018' 


I 348 


3 282^ 


t 8420 8-021048' 


r 9217 


478£ 


7364 55 




6 2435 


5 567. 


) 494' 


r 8-009048: 


{ 24938-033116C 


668S 


9216154 




7 469^ 


I 7865 


I 707( 


) 25451 4496 


] 312C 


8586 


8-056106' 


^53 




8 694E 


) 7-984004' 


r 919] 


460f 


650] 


506£ 


8-0450487 


291' 


^52 




9 919^ 


5 223] 


7-9971311 


666i 


) 8504 


701E 


2385 


476' 


^51 




10 7.971144C 


4414 


343C 


872f 


8-022050C 


8967 


4282 


661E 


50 




11 369E 


6596 


5546 


8-0100782 


2507 


S-0340914 


6178 


846- 


49 




15 5947 


8777 


7666 


284C 


4507 


286C 


8074 


8-057031C 


48 




13 819-^ 


7-9850957 


9782 


4896 


6507 


4806 


9968 


2156 


47 




14 7.9720441 


313£ 


7-9981897 


6951 


8505 


6750 


8-0461862 


4002 


46 




15 2686 


5313 


4011 


9005 


8-0230502 


8694 


3755 


5846 


45 




16 4930 


7490 


6124 


8-0111058 


2499 


8-0350637 


5647 


769C 


44 




17 7173 


9665 


8236 


3110 


4494 


2579 


7538 


9534 


43 




18 9414 


7-9861839 


7-9990346 


5161 


6489 


4520 


9429 


8-0581376 


42 




19 7-9731655 


4013 


2456 7211 


8483 


6460 


8-0471318 


3217 


41 




20 3894 


6185 


4565 9260 


8-0240475 


8400 


3207 


5058 


40 




21 6132 


8356 


6673 


8-0121308 


2467 


8-0360338 


5095 


6898 


39 




22 8369 


7-9870526 


8780 


3356 


4458 


2276 


6982 


8737 


38 




23 7-9740605 


2695 


8-0000886 


5402 


6448 


4213 


8869 


80590576 


37 




24 2840 


4862 


2991 


7447 


8437 


6149 


8-0480754 


2414 


36 




25 5073 


7029 


5094 


9492 


8-0250426 


8084 


2639 


4250 


35 




26 7306 

27 9537 

28 7-9751767 


9195 


7197 


8-0131535 


2413 


8-0370018 


4523 


6087 


34 




7-9881359 


9299 


3578 


4399 


1951 


6406 


7922 


33 




3523 


8-0011400 


5619 


6385 


3884 


8288 


9756 


32 




j29 3996 


5685 


3499 


7660 


8369 


5815 


8-0490169 


8-0601590 


31 




30 6224 


7847 


5598 


9699 


8-0260353 


7746 


2050 


3423 


30 




31 8451 


7-9890007 


7696 


8-0141438 


2336 


9676 


3930 


5255 


29 




32 7-9760676 


2166 


9792 


3775 


4318 


8-0381605 


5809 


7087 


28 




33 2901 


4324 


8-0021888 


5812 


6299 


3533 


7687 


8918 


27 




34 5124 


6481 


3983 


7848 


8279 


5461 


9564 


8-0610748 


26 




35 7346 


8637 


6076 


9883 


8-0270258 


7387 


8-0501441 


2577 


25 




36 9567 


7-9900792 


8169 


8-0151916 


2236 


9313 


3317 


4405 


24 




37 7-9771787 


2946 


8 0030260 


3949 


4213 


80391238 


5192 


6233 


23 




38 4006 


5099 


2351 


5981 


6190 


3162 


7066 


8060 


22 




39 6224 


7251 


4441 


■8012 


8166 


5085 


8939 


9886 


21 




40 8440 


9401 


6529 


3-0160042 


8-02S0140 


7007 


8-0510812 


8-0621711 


20 




41 7-9780655 


7-9911551 


8617 


2071 


2114 


8928 


2683 


3536 


19 




42 2870 


3699 


8-0040703 


4099 


4087 


8-0400849 


4554 


5359 


18 




43 5083 


6847 


2789 


6127 


6059 


2768 


6424 


7182 


17 




44 7295 


7993 


4874 


8153 


8030 


4687 


8294 


9005 


16 




45 9506 


7-9920138 


6957 


3-0170178 


8-0290000 


6605 


8-0520162 


3-0630826 


15 




46 7-9791715 


2283 


9040 


2203 


1969 


8522 


2030 


2647 


14 




47 3924 


4426 


3-0051121 


4226 


3938 


3-0410439 


3897 


4467 


13 




48 6131 


6568 


3202 


6248 


5905 


2354 


5763 


6286 


12 




49 8338 


8709 


5282 


8270 


7872 


4269 


7628 


8104 


11 




50 7-9800543 


r-9930849 


7360 


30180291 


9838 


6183 


9493 


9922 


10 




51 2747 
152 4950 


2988 


9438 


2310 


3-0301802 


8096 


3-0531356 i 


3-0641739 


9 




5126 


30061514 


4329 


3766 


3-0420008 


3219 


3555 


8 




|53 7152 


7263 


3590 


6347 


5729 


1919 


5081 


5371 


7 




54 9353 


9399 


5665 


8364 


7692 


3829 


6943 


7185 


6 




55 7-9811552' 


'-•9941534 


7738? 


30190379 


9653 


5739 


8803 


8999 


5 




56 3751 


3667 


9811 


2394 5 


5-0311613 


7648 J 


3-0540663 8 


3-0650812 


4 




57 5948 


5800 


?-0071883 


4408 


3573 


9555 


2522 


2625 


3 




58 8145 


7932 


3953 


6422 


553H 


J-0431462 


4380 


4436 


2 




59 7-9820340 7 


'-9950062 


6023 


8434 


7489 


3369 


6237 


6247 


1 




60 2534 2192| 


8092 s 


•0200445 


9446 


5274 


8094 


8057 







^'1 27' 26' 1 


25' 


24' 


23' 


22' 


21' 


20' 


'M 




LOG. COTAN. 89°. j| 





74 LOG. SINE 0=^. [Table ] 


^ 


'/ 


40' 


41' 


42' 


43' 


44' 


45' 


46' 


47' 


1 

" 





8-0657763 


8-0764997 


8-0869646 


3-G971832 


3-1071669 


3-1169262 


1-1264710 


3-1358104 


60 


1 


9572 


6762 


3-0871369 


3515 


3314 


3-1170870 


6283 


9644 


59 


2 


8-0661381 


8526 


3091 


5198 


4958 


2478 


7856 


3-1361183 


58 


3 


3188 


8-0770290 


4813 


6879 


6601 


4085 


9428 


2722 


57 


4 


4995 


2052 


6534 


8560 


8244 


5691 


3-1270999 


4260 


56 


5 


6801 


3815 


8254 


3-0980240 


9886 


7297 


2570 


5797 


55 


6 


8606 


5576 


9974 


1920 


3-1081528 


8902 


4140 


7334 


54 


7 


8-C670411 


7337 


8-0881692 


3599 


3169 


8-1180507 


5710 


8871 


53 


8 


2215 


9097 


3411 


5277 


4809 


2111 


7279 


8-1370407 


52 


9 


4018 


8-0780856 


5128 


6955 


6449 


3714 


8848 


1942 


51 


10 


5820 


2614 


6845 


8632 


8088 


5317 


81280416 


3477 


50 


11 


7622 


4372 


8561 


3-0990309 


9726 


6919 


1983 


5011 


49 


12 


9423 


6129 


8-0890277 


1984 


8-1091364 


8520 


3550 


6545 


48 


13 


8-0681223 


7886 


1991 


3659 


3001 


8-119012L 


5117 


8078 


47 


14 


3022 


9641 


3706 


5334 


4638 


1722 


6682 


9610 


46 


15 


4821 


8-0791396 


5419 


7008 


6274 


3322 


8248 


8-1381143 


45 


16 


6619 


3151 


7132 


8681 


7909 


4921 


9812 


2674 


44 


17 


8416 


4904 


8844 


8-1000353 


9544 


6519 


8-1291376 


4205 


43 


18 


8-0690212 


6657 


8-0900555 


2025 


8-1101178 


8118 


2940 


5736 


42 


19 


2008 


8409 


2266 


3697 


2812 


9715 


4503 


7265 


41 


20 


3803 


8-0800161 


3976 


5367 


4445 


8-1201312 


6065 


8795 


40 


21 


5597 


1912 


5685 


7037 


6077 


2908 


7627 


8-1390324 


39 


22 


7390 


3662 


7394 


8706 


7709 


4504 


9188 


1855 


38 


23 


9183 


5411 


9102 


8-1010375 


9340 


6099 


8-1300749 


338C 


37 


24 


8-0700975 


7160 


8-0910810 


2043 


8-1110970 


7693 


2309 


4907 


36 


25 


2766 


8908 


2516 


3710 


2600 


9287 


3669 


6434 


35 


26 


4557 


80810655 


4222 


5377 


4229 


8-1210881 


5428 


796C 


34 


27 


6346 


2401 


5928 


7043 


5858 


2474 


6966 


948£ 


33 


28 


8135 


4147 


7632 


8709 


7486 


4066 


8544 


8-1401011 


32 


29 


9923 


5892 


9336 


8-1020374 


9113 


5657 


8-1310101 


2531 


31 


30 


8-0711711 


7637 


8-0921040 


2038 


8-1120740 


7248 


1658 


405C 


)30 


31 


3498 


9380 


2743 


3701 


2366 


8839 


3215 


558: 


)29 


|32 


5284 


8-0821123 


4445 


5364 


3992 


8-1220429 


4770 


im 


>28 


33 


7069 


2866 


6146 


7027 


5617 


2018 


6325 


8625 


27 


34 


8854 


4607 


7847 


8688 


7241 


3607 


7880 


8-14l0l5( 


)26 


35 


8-0720637 


6348 


9547 


8-1030349 


8865 


5195 


9434 


1671 


25 


36 


2421 


8088 


8 0931246 


2010 


8-1130488 


6782 


8-1320987 


319^ 


>24 


37 


4203 


9828 


2945 


3669 


2110 


8369 


2540 


471^ 


123 


38 


5985 


8-0831567 


4643 


5328 


3732 


9956 


4093 


623^ 


122 


39 


7765 


3305 


6340 


6987 


5354 


8-1231541 


5644 


7751 


L21 


40 


9546 


5042 


8037 


8645 


6974 


3127 


7196 


927( 


)20 


41 


8-0731325 


6779 


9733 


8-1040302 


8595 


4711 


8746 


8-142078J 


n9 


42 


3104 


8515 


8-0941428 


1959 


8-1140214 


6295 


81330296 


230( 


)18 


43 


4882 


8-0840251 


3123 


3615 


1833 


7879 


1846 


382: 


m 


44 


6659 


1985 


4817 


5270 


3451 


9462 


3395 


533J 


)16 


45 


8436 


3719 


6510 


6925 


5069 


8-1241044 


4943 


685J 


)15 


46 


8-0740211 


5452 


8203 


6579 


6686 


2626 


6491 


837 


14 


47 


1986 


7185 


9895 


8-1050232 


8302 


4207 


8039 


988( 


513 


48 


3761 


8917 


8-0951587 


1885 


9918 


5787 


9586 


8-1431 40( 


)12 


49 


5534 


8-0850648 


3277 


3537 


8-1151534 


7367 


8-1341132 


291^ 


111 


50 


7307 


2379 


4968 


5188 


3148 


8947 


2678 


442' 


no 


51 


9080 


4109 


6657 


6839 


4762 


8-1250526 


4223 


594( 


) 9 


52 


8-0750851 


5838 


8346 


8490 


6376 


2104 


5767 


745: 


i 8 


53 


2622 


7566 


8-0960034 


8-1060139 


7989 


3682 


7311 


896^ 


[ 7 


54 


4392 


9294 


1721 


1788 


9601 


5259 


8855 


8-144047( 


] 6 


65 


6161 


8-0861021 


3408 


3437 


8-1161213 


6836 


8-1350398 


198' 


r 5 


66 


7930 


2747 


5094 


5085 


2824 


8412 


1940 


349' 


4 


57 


9698 


4473 


6780 


6732 


4434 


9987 


3482 


500f 


3 


58 


8-0761465 


6198 


8465 


8378 


6044 


8-1261562 


5023 


65ie 


2 


59 


3231 7922 


8-0970149 


8-1070024 


7654 


3136 


6564 


8024 


1 


60 


4997 9646 


1832 


1669 


9262 


4710 


8104 


9532 





// 


19' 18' 


17' 


16' 


15' 


14' 


13' 


12' 


nl 


LOG. COSINE 89°. Jj 



Table ii.] log. tan. 0°. 75 || 


" 


40' 


4r 


42' 


43' 


44' 


45' 


46' 


47' 1" il 


C 


8-0658057 


80765306 


80869970 


8-0972172 


8-1072025 


81169634 


81265099 


81358510 


60 


1 


9866 


7071 


8-0871693 


3855 


367C 


81171243 


6672 


81360050 


59 




8-0661675 


8835 


3416 


5538 


5314 


2851 


8245 


1590 


68 


3 


3483 


8-0770599 


5138 


7220 


695£ 


4458 


9817 


3129 


57 


4 


529C 


2362 


6859 


8901 


8601 


6064 


8-1271389 


4667 


56 


5 


7096 


4125 


8579 


8-0980582 


8-1080243 


7670 


2960 


6205 


55 


6 


8902 


5886 


8-0880299 


2261 


1885 


9276 


4531 


7742 


54 


7 


8-0670707 


7647 


2018 


3941 


3526 


8-1180881 


6101 


9279 


53 


8 


2511 


9407 


3737 


5619 


5167 


2485 


7670 


8-1370815 


52 


9 


4314 


80781167 


5455 


7297 


6807 


4088 


9239 


2350 


51 


10 


6117 


2926 


7172 


8975 


8446 


5691 


8-1280807 


3886 


50 


11 


7919 


4684 


8888 


S-0990651 


8-1090085 


7294 


2375 


5420 


49 


12 


9720 


6441 


8-0890604 


2327 


1723 


8896 


3942 


6954 


48 


13 


8-0681520 


8198 


2319 


4003 


3361 


8-1190497 


5509 


8488 


47 


14 


3320 


9954 


4033 


5677 


499E 


2098 


7075 


8-1380020 


46 


15 


5118 


8-0791709 


5747 


7351 


6634 


3698 


8641 


1553 


45 


16 


6917 


3464 


7460 


9025 


826G 


5297 


8-1290206 


3085 


44 


17 


8714 


5218 


9172 


8-1000698 


9904 


6896 


1770 


4616 


43 


18 


8-0690511 


6971 


8-0900884 


2374 


8-n0153£ 


8495 


3334 


6147 


42 


19 


2306 


6723 


2595 


4041 


3173 


8-1200092 


4897 


7677 


41 


20 


4102 


8-0800475 


4305 


5712 


4806 


1689 


6460 


9207 


40 


21 


5896 


2226 


6015 


7382 


6438 


3286 


8022 


8-1390736 


39' 


22 


7690 


3976 


7724 


9052 


8070 


4882 


9583 


2264 


38 


23 


9483 


5726 


9432 


8-1010721 


9702 


6477 


8-1301144 


3792 


37 


24 


8-0701275 


7475 


809 LI 140 


2389 


8-1111332 


8072 


2705 


5320 


36 


25 


3066 


9223 


2847 


4057 


2962 


9666 


4265 


6847 


35 


26 


4857 


80810970 


4553 


5724 


4592 


8-1211260 


5824 


8373 


34 


27 


6647 


2717 


6259 


7390 


6221 


2853 


7383 


9899 


33 


28 


8436 


4463 


7964 


9056 


7849 


4446 


8941 


8-1401425 


32 


29 


8-0710225 


6208 


9668 


8-1020721 


9477 


6037 


8-1310498 


2949 


31 


30 


2012 


7953 


80921372 


2386 


81121104 


7629 


2056 


4474 


30 


31 


3799 


9697 


3075 


4049 


2730 


9219 


3612 


5997 


29 


32 


5586 


8-0821440 


4777 


5713 


4356 


8-1220810 


5168 


7521 


28 


33 


7371 


3183 


6479 


7375 


5981 


2399 


6723 


9043 


27 


134 


9156 


4925 


8180 


9037 


7606 


3988 


8278 


8-1410566 


26 


35 


8-0720940 


6666 


9880 


8-1030698 


9230 


5577 


9833 


2087 


25 


36 


2723 


8406 


8-0931579 


2359 


8-1130853 


7164 


8-1321386 


3608 


24 


37 


4506 


8-0830146 


3278 


4019 


2476 


8752 


2940 


5129 


23 


l38 


6288 


1885 


4977 


5678 


4098 


8-1230338 


4492 


6649 


22 


139 


8069 


3624 


6674 


7337 


5720 


1924 


6044 


8168 


21 


140 


9850 


5361 


8371 


8995 


7341 


3510 


7596 


9687 


20 


41 


8-0731629 


7098 


8-0940068 


8-1040653 


8961 


5095 


9147 


8-1421206 


19 


42 


3408 


8835 


1763 


2309 


8-1140581 


6679 


8-1330697 


2724 


18 


43 


5186 


80840570 


3458 


3966 


2200 


8263 


2247 


4241 


17 


44 


6964 


2305 


5153 


5621 


3819 


9846 


3796 


5758 


16 


45 


8741 


4039 


6846 


7276 


5437 


8-1241429 


5345 


7274 


15 


46 


8-0740517 


5773 


8539 


8931 


7054 


3011 


6893 


8790 


14 


47 


2292 


7506 


8 0950232 


8-1050584 


8671 


4592 


8441 


8-1430305 


13 


48 


4067 


9238 


1923 


2237 


8-1150287 


6173 


9988 


1820 


12 


49 


5841 


8-0850969 


3614 


3890 


1903 


7753 


8-1341535 


3334 


11 


50 


7614 


2700 


5305 


5542 


3518 


9333 


3081 


4848 


10 


51 


9386 


4430 


6994 


7193 


5132 


8-1250912 


4626 


6361 


9 


52 


8-0751158 


6160 8683 


8843 


6746 


2491 


6171 


7874 


8 


53 


2929 


78888-0960372 


8-1060493 


8359 


4069 


7715 


9386 


7 


54 


4699 


9616 


2060 


2142 


9972 


5646 


9259 


8-1440897 


6 


55 


6469 


8-0861344 


3747 


3791 


3-1161584 


7223 


3-1350802 


2408 


5 


56 


8238 


3070 


5433 


5439 


3195 


8799 


2345 


3919 


4 


57 


3-0760006 


4796 


7119 


7087 


4806 


8-1260375 


3887 


5429 


3 


5.8 


1773 


6522 


8804 


8733 


6416 


1950 


5429 


6938 


2 


59 


3540 


8246 


3-0970488 8-1070380 


8025 


3525 


6970 


8447 


1 


60 


5306 


9970 


2172 2025 


9634 


5099 8510 


9956 





// 


19^ 18' 1 


17' 16' 15' i 


14' 1 13' 


12' 


// 


LOG. COTAN. 89°. Jj 



76 


LOG. SINE Oo. {Tablt II. ll 


o 


48' 


49' 


50' 


51' 


52' 


53' 


54' 


55' " II 





8-1449532 


S-1539075 


8-1626808 


8-1712604 


8-1797129 


8-1879848 


3-1961020 


8-2040703 


60 


1 


8-1451040 


3-1540552 


8255 


4223 


8521 


S-1881213 


2360 


2019 


59 


2 


2547 


2028 


9702 


5641 


9912 


2578 


3700 


3334 


58 


3 


4054 


3504 


3-1631 149 


7059 


8-1801303 


3943 


5039 


4649 


57 


4 


5560 


4979 


2594 


8477 


2693 


5307 


6378 


5963 


56 


5 


7065 


6454 


4040 


9894 


4083 


6670 


7717 


7277 


55 


6 


8570 


7928 


5485 


8-1721310 


5472 


8034 


9055 


8591 


54 


7 


8-1460075 


9402 


6929 


2726 


6861 


9397 


8-1970392 


9905 


53 


8 


1579 


8-1550876 


8373 


4142 


8250 


8-1890759 


1729 


8-2051218 


52 


9 


3082 


2348 


9817 


5557 


9638 


2121 


3066 


2530 


51 


10 


4585 


3821 


8-1641259 


6972 


8-1811025 


3482 


4403 


3842 


50 


11 


6087 


5293 


2702 


8386 


2413 


4843 


5739 


5154 


49 


12 


7589 


6764 


4144 


9800 


3799 


6204 


7074 


6465 


48 


13 


9091 


8235 


5586 


8-1731214 


5186 


7564 


8409 


7776 


47 


14 


8-1470591 


9705 


7027 


2627 


6571 


8924 


9744 


9087 


46 


15 


2092 


S-1561175 


8467 


4039 


7957 


8-1900284 


8-1981078 


8-2060397 


45 


16 


3592 


2644 


9907 


5451 


9342 


1643 


2412 


1707 


44 


17 


5091 


4113 


3-1651347 


6863 


8-1820726 


3001 


3746 


3016 


43 


18 


6590 


5582 


2786 


8274 


2111 


4359 


5079 


4325 


42 


19 


8088 


7049 


4225 


9684 


3494 


5717 


6412 


5634 


41 


20 


9586 


8517 


5663 


S-] 741094 


4877 


7074 


7744 


6942 


40 


21 


8-1481083 


9984 


7101 


2504 


6260 


8431 


9076 


8250 


39 


22 


2579 


8-1571450 


8538 


3913 


7643 


9788 


9-1990407 


9557 


38 


23 


4076 


2916 


9975 


5322 


9024 


8-1911144 


1738 


8-2070864 


37 


24 


5571 


4381 


8-1661411 


6731 


8-1830406 


2499 


3069 


2171 


36 


25 


7066 


5846 


2847 


8138 


1787 


3854 


4399 


3477 


35 


26 


8561 


7310 


4282 


9546 


3167 


5209 


5729 


4783 


34 


27 


8-1490055 


8774 


5717 


8-1750953 


4548 


6563 


7058 


6088 


33 


28 


1549 


8-1580238 


7151 


2359 


5927 


7917 


8387 


7393 


32 


29 


3042 


1701 


8585 


3765 


7307 


9271 


9716 


8698 


31 


30 


4534 


3163 


81670019 


5171 


8685 


8-1920624 


8-2001044 


6-2080002 


30 


31 


6027 


4625 


1452 


6576 


8-1840064 


1976 


2372 


1306 


29 


32 


7518 


6086 


2884 


7981 


1442 


3329 


3699 


2610 


28 


33 


9009 


7547 


4316 


9385 


2819 


4680 


5026 


3913 


27 


34 


8-1500500 


9008 


5748 


8-1760789 


4196 


6032 


6353 


5216 


26 


35 


1990 


8-1590468 


7179 


2192 


5573 


7383 


7679 


6518 


25- 


36 


3479 


1927 


8610 


3595 


6949 


" 8733 


9005 


7820 


24 


37 


4968 


3386 


8-1680040 


4998 


8325 


8-1930083 


8-2010330 


9121 


23 


38 


6457 


4845 


1469 


6400 


9700 


1433 


1655 


8 2090422 


22 


39 


7945 


6303 


2899 


7801 


8-1851075 


2782 


2980 


1723 


21 


40 


9432 


7760 


4327 


9202 


2450 


4131 


4304 


3024 


20 


41 


8-1510919 


9217 


5756 


8-1770603 


3824 


5479 


5628 


4324 


19 


42 


2406 


8-1600674 


7183 


2003 


5197 


6827 


6951 


5623 


18 


43 


3891 


2130 


8611 


3403 


6570 


8175 


8274 


6922 


17 


44 


5377 


3585 


8-1690038 


4802 


7943 


9522 


9597 


8221 


16 


45 


6862 


5040 


1464 


6201 


9315 


8-1940869 


8-2020919 


9520 


15 


46 


8346 


6495 


2890 


7599 


8-1860687 


2215 


2241 


8-2100818 


14 


47 


9830 


7949 


4315 


8997 


2059 


3561 


3562 


2115 


13 


48 


81521314 


9403 


5740 


8-1780394 


3430 


4907 


4883 


3412 


12 


49 


2796 


8-1610856 


7165 


1791 


4800 


6252 


6203 


4709 


11 


50 


4279 


2308 


8589 


3188 


6170 


7596 


7523 


6006 


10 


51 


5761 


3761 


8-1700012 


4584 


7540 


8941 


8843 


7302 


9 


52 


7242 


5212 


1435 


5980 


8909 


8-1950284 


8-2030163 


8598 


8 


53 


8723 


6663 


2858 


7375 


8-1870278 


1628 


1481 


9893 


7 


54 


8-1530203 


8114 


4280 


8770 


1646 


2971 


2800 


8-2111188 


6 


55 


1683 


9564 


5702 


8-1790164 


3014 


4313 


4118 


2482 


5 


56 


3163 


8-1621014 


7123 


1558 


4382 


5656 


5436 


3777 


4 


57 


4641 


2463 


8544 


2951 


5749 


6997 


6753 


5070 


3 


58 


6120 


3912 


9964 


4344 


7116 


8339 


8070 


6364 


2 


59 


7598 


5360 


8-1711384 


5737 


8482 


9680 


9387 


7657 


1 


60 


9075 


6808 


2804 


7129 


9848 


8-1961020 


8-2040703 


8949 





// 


11' 1 10' 


9' 


8' 


7' 1 6' 


5' 


4' 


// 


•LOG. COSINE. f| 



1 Table ii.] log. tan. Oo. 77 | 


/' 


1 48' 


49' 


50' 


51' 


52' 


53' 


54' 


55' 


'/ 





^•1449956 


S1539516 


3-1627267 


8-1713282 


8-1797626 


8-1880364 


8-1961556 


8-2041259 


60 


1 


D-1451464 


S-i&l0993 


8715 


4701 


9018 


1730 


2896 


2575 


59 


2 


2971 


2470 


3-1630162 


6120 


8-1800409 


3095 


4236 


3890 


58 


3 


4478 


3946 


1609 


7538 


1800 


4460 


5576 


5206 


57 


4 


5984 


5422 


3055 


8956 


3191 


5824 


6915 


6521 


56 


5 


7490 


6897 


4501 


8-1720373 


4581 


7188 


8254 


7835 


55 


6 


8995 


8371 


5946 


1790 


5971 


8552 


9592 


9149 


54 


7 


S-1460500 


9846 


7391 


3207 


7360 


9915 


3-1970930 


8 2950463 


53 


8 


2004 


S-1551319 


6835 


4623 


8749 


8-1891278 


2268 


1776 


52 


9 


3508 


2792 


3-J640279 


6038 


8-1810137 


2640 


3605 


3089 


51 


10 


5011 


4265 


1722 


7453 


1525 


4002 


4942 


4401 


50 


11 


6514 


5737 


3165 


8868 


2913 


5363 


6278 


5714 


49 


12 


8016 


7209 


4607 


8-1730282 


4300 


6724 


7614 


7025 


48 


13 


9518 


8680 


6049 


1696 


5687 


8085 


8949 


8337 


47 


14 


S-1471019 


8-1560151 


7490 


3109 


7073 


9445 


8-1980284 


9647 


46 


15 


2520 


1621 


8931 


4522 


8459 


9-1900805 


1619 


8-2060958 


45 


16 


4020 


3090 


3-1650372 


5934 


9844 


2164 


2953 


2268 


44 


17 


5519 


4559 


1812 


7346 


8-1821229 


3523 


4287 


3578 


43 


IS 


7018 


6028 


3251 


8757 


2613 


4881 


5621 


4887 


42 


19 


8517 


7496 


4690 


8-1740168 


3997 


6239 


6954 


6196 


41 


20 


81480015 


8964 


6128 


1579 


5381 


7597 


8286 


7505 


40 


21 


1512 


8-1570431 


7566 


2989 


6764 


8954 


9619 


8813 


39 


22 


3009 


1898 


9004 


4398 


8146 


8-1910311 


8-1990950 


8-2070120 


38 


23 


4506 


3364 


3-1660441 


5807 


9529 


1667 


2282 


1428 


37 


24 


6002 


4830 


1878 


7216 


8-1830910 


3023 


3613 


2735 


36 


25 


7497 


6295 


3314 


8624 


2292 


4379 


4943 


4041 


35 


26 


' 8992 


7759 


4749 


8-1750032 


3673 


5734 


6273 


5348 


34 


27 


8-1490487 


9224 


6185 


1439 


5053 


7088 


7603 


6653 


33 


28 


1980 


8-1580687 


7619 


2846 


6433 


8442 


8933 


7959 


32 


29 


3474 


2151 


9054 


4252 


7813 


9796 


8-2000262 


9264 


31 


30 


4967 


3613 


8-1670487 


5658 


9192 


8-1921150 


1590 


8-2080568 


30 


31 


6459 


5076 


1921 


7064 


8-1840571 


2503 


2918 


1873 


29 


32 


7951 


6537 


3353 


8469 


1949 


3855 


4246 


3176 


28 


33 


9442 


7999 


4786 


9873 


3327 


5207 


5573 


4480 


27 


34 


8-1500933 


9459 


6218 


8-1761278 


4704 


6559 


6900 


5783 


26 


35 


2423 


8-1590920 


7649 


2681 


6081 


7910 


8227 


7086 


25 


36 


3913 


2379 


9080 


4084 


7458 


9261 


9553 


8388 


24 


37 


5402 


3839 


3-1680510 


5487 


8834 


8-1930611 


8-2010879 


9690 


23 


38 


6891 


5297 


1940 


6889 


8-1850209 


1961 


2204 


8-2090991 


22 


39 


8380 


6756 


3370 


8291 


1585 


3311 


3529 


2292 


21 


40 


9867 


8213 


4799 


9693 


2959 


4660 


4853 


3593 


20 


41 


8-1511355 


9671 


6228 


8-1771094 


4334 


6009 


6177 


4893 


19 


42 


2841 


8-1601128 


7656 


2494 


5708 


7357 


7501 


6193 


18 


43 


4328 


2584 


9083 


3894 


7081 


8705 


8824 


7493 


17 


44 


5813 


4040 


8-1690510 


5294 


8454 


8-1940053 


8-2020147 


8792 


16 


45 


7299 


5495 


1937 


6693 


9827 


1400 


147C 


8-2100091 


15 


46 


8783 


6950 


3363 


8091 


8-1861199 


2746 


2792 


1389 


14 


47 


8-1520267 


8404 


4789 


9490 


2571 


4093 


4113 


2687 


13 


48 


1751 


9858 


6214 


8-1780887 


3942 


5439 


5435 


3985 


12 


49 


3234 


8-1611312 


7639 


2285 


5313 


6784 


6756 


5282 


11 


50 


4717 


2765 


9064 


3682 


6683 


8129 


8076 


6579 


10 


51 


6199 


4217 


8-1700487 


5078 


8053 


9473 


9396 


7875 


9 


52 


7681 


5669 


1911 


6474 


9423 


8-1950818 


8-2030716 


9171 


8 


53 


9162 


7121 


3334 


7870 


8-1870792 


2161 


2035 


8-2110467 


7 


54 


8-1530643 


8572 


4756 


9265 


2161 


3505 


3354 


1762 


6 


55 


2123 


8-1620022 


6178 


8-1790659 


3529 


4848 


4672 


3057 


5 


56 


3603 


1472 


7600 


2054 


4897 


6190 


5990 


4351 


4. 


57 


5082 


2922 


9021 


3447 


6264 


7532 


7308 


5646 


3 


58 


6560 


4371 


8-1710442 


4841 


7631 


8874 


8625 


6939 


2. 


59 


8038 


5819 


1862 


6233 


8998 


8-1960215 


9942 


8233 


1 


60 


9516 


7267 


3282 


7626 


8-1880364 


1556 


8-2041259 


9526 





// 


ir. 


10' 


9' 


8' 


r 


6' 


5' 


4' 


// 


LOG. COTAN. Sa^'. 11 



7* 



78 LOG. SINE 0°. LOG. SINE 1°. [TuMe II. I 


// 


56' 


57' 


58' 


59' 


0' 


1' 


2' 1 


3' 


" 





8-2118949 


8-2195811 


8-2271335 


S-2345568 


3-2418553 


8-2490332 


8-2560943 


8-2630424 


60 


1 


8'2120242 


7080 


2583 


6795 


9759 


1518 


2120 


1572 


59 


2 


1533 


8349 


3830 


8021 


8-2420965 


2704 


3277 


2721 


58 


1 3 


2825 


9618 


5077 


9247 


2170 


3890 


4443 


3869 


57 


1 4 


4116 


8-2200887 


6324 


82350472 


3376 


5075 


5609 


5016 


56 


5 


5407 


2155 


7570 


1697 


4580 


6260 


6775 


6164 


55 


6 


6697 


3423 


8816 


2922 


5785 


7445 


7941 


7311 


54 


7 


7987 


4690 


8-2-280061 


4147 


6989 


8629 


9106 


8458 


53 


6 


9277 


5957 


1306 


5371 


8192 


9813 


8-2570271 


9604 


52 


9 


8-2130566 


7223 


2551 


6594 


9396 


3-2500997 


1436 


8-2640750 


51 


10 


1854 


8490 


3796 


7818 


8-2430599 


2180 


2600 


1896 


50 


11 


3143 


9756 


5040 


9041 


1802 


3363 


3764 


3042 


49 


12 


4431 


8-2211021 


6284 


8-2360264 


3004 


4546 


4928 


4187 


48 


13 


5719 


2286 


7527 


1486 


4206 


5728 


. 6091 


5332 


47 


14 


7006 


3551 


8770 


2708 


5408 


6911 


7255 


6477 


46 


15 


8293 


4815 


8-2290013 


3930 


6609 


8092 


8417 


7621 


45 


16 


9579 


6079 


1255 


5151 


7810 


9274 


9580 


8766 


44 


17 


8-2140865 


7343 


2497 


6372 


9011 


8-2510455 


8-2580742 


9909 


43 


18 


2151 


8606 


3739 


7593 


8-2440212 


1636 


1904 


8-2651053 


42 


19 


3436 


9869 


4980 


8813 


1412 


2816 


3065 


2196 


41 


20 


4721 


8-2221132 


6221 


8-2370033 


2611 


3996 


4227 


3339 


40 


21 


6006 


2394 


7461 


1253 


3811 


5176 


5388 


4482 


39 


22 


7290 


3636 


8701 


2472 


5010 


6356 


6548 


5624 


38 


23 


8574 


4917 


9941 


3691 


6209 


7535 


7709 


6766 


37 


24 


9857 


6178 


8-2301181 


4910 


7407 


8714 


8869 


7908 


36 


25 


8-2151140 


7439 


2420 


6128 


8605 


9893 


8-2590028 


9049 


35 


26 


2423 


8699 


3659 


7346 


9803 


8-2521071 


1188 


8-2660190 


34 


27 


3705 


9959 


4897 


8563 


8-2451000 


2249 


2347 


1331 


33 


28 


4987 


8-2231219 


6135 


9781 


2198 


3426 


3505 


2471 


32 


29 


6269 


2478 


7373 


8-2380997 


3394 


4604 


4664 


3612 


31 


30 


7550 


3737 


8610 


2214 


4591 


5781 


5822 


4751 


30 


31 


8831 


4996 


9847 


3430 


5787 


6957 


6980 


5891 


29 


32 


8-2160111 


6254 


8-2311084 


4646 


6983 


8134 


8137 


7030 


28 


33 


1391 


7512 


2320 


5862 


8178 


9310 


9295 


8169 


27 


34 


2671 


8769 


3556 


7077 


9373 


8-2530485 


8-2600452 


9308 


26 


35 


3950 


8-2240026 


4792 


8292 


8-2460568 


1661 


1608 


8-2670446 


25 


36 


5229 


1283 


6027 


9506 


1762 


2836 


2764 


1585 


24 


37 


6508 


2539 


7262 


8-2390720 


2957 


4011 


3920 


2722 


23 


38 


7786 


3795 


8496 


1934 


4150 


5185 


5076 


3860 


22 


39 


9064 


5051 


9731 


3148 


5344 


6359 


6232 


4997 


21 


40 


8-2170341 


6306 


8-2320965 


4361 


6537 


7533 


7387 


6134 


20 


41 


1618 


7561 


2198 


5574 


7730 


8706 


8541 


7271 


19 


42 


2895 


8815 


3431 


6786 


8922 


9880 


9696 


8407 


18 


43 


4171 


8-2250070 


4664 


7998 


8-2470115 


8-2541052 


8-2610850 


9543 


17 


44 


5447 


1323 


5896 


9210 


1306 


2225 


2004 


8-2680679 


16 


45 


6723 


2577 


7128 


8-2400422 


2498 


3397 


3157 


1814 


15 


46 


7998 


3830 


8360 


1633 


3689 


4569 


4311 


2949 


14 


47 


9273 


5083 


9592 


2844 


4880 


5741 


5463 


4084 


13 


48 


8-2180547 


6335 


8-2330823 


4054 


6071 


6912 


6616 


5219 


12 


49 


1821 


7587 


2053 


5264 


7261 


8083 


7768 


6353 


11 


50 


3095 


8839 


3284 


6474 


8451 


9254 


8920 


7487 


10 


51 


4368 


8-2260090 


4514 


7683 


9640 


8-2550424 


8-2620072 


8620 


9 


52 


5641 


1341 


5743 


8892 


3-2480829 


1594 


1223 


9754 


8 


53 


6913 


2591 


6973 


8-2410101 


2018 


2764 


2375 


8-2690887 


7 


54 


8186 


3841 


8202 


1310 


3207 


3933 


3525 


2020 


6 


55 


9457 


5091 


9430 


2518 


4395 


5102 


4676 


3152 


5 


56 


8-2190729 


6341 


8-2340659 


3725 


5583 


6271 


5826 


4284 


4 


57 


2000 


7590 


1886 


4933 


6771 


7439 


6976 


5416 


3 


58 


3270 


8839 


3114 


6140 


7958 


8607 


8125 


6548 


2 


59 


4541 


8-2270087 


4341 


7347 


9145 


9775 


9275 


7679 


1 


60 


5811 


1335 


5568 


8553 


3-2490332 


8-2560943 


8-2630424 


8810 





// 


3' 


2' 


1' 


0' 


59' 


58' 


57' 


56' 


// 


|| LOG. COSINE 89°. LOG. COSINE 88°. )| 



Table ii.] log. tan. 0°. log. tan. °1. 


^1 


' 


1 56' 


57' 


58' 


59' 


0' 


1' 


2' 


3' 


// 


( 


) S-2ll952e 


S-219640£ 


S-2271953 


8-2346208 


8-2419215 


8-2491015 


8-2561649 


8-2631153 


60 


! ] 


8'2120S1£ 


7676 


3201 


7435 


8-2420421 


2202 


2817 


2302 


59 


1 I 


> 2110 


6947 


4449 


8661 


1627 


3388 


3984 


345] 


58 




3402 


5-2200216 


5696 


9887 


2833 


4574 


5151 


4599 


57 


\ 


4694 


1485 


6943 


8-2351113 


4038 


5760 


6317 


5747 


56 


5 


5985 


2754 


8190 


2339 


5244 


6946 


7484 


6895 


55 


I 6 


7275 


4022 


9436 


3564 


6448 


8131 


8650 


8043 


54 


1 7 


8566 


5289 


8-2280682 


4789 


7653 


9315 


9815 


9190 


53 


8 


9855 


6557 


1927 


6013 


8857 


8-2500500 


3-2570981 


8-2640337 


52 


9 


8-2131145 


7824 


3173 


7237 


S-2430061 


1684 


2146 


1483 


51 


10 


2434 


9090 


4417 


8461 


1264 


2868 


3310 


2630 


50 


11 


3723 


S-2210356 


5662 


9684 


2467 


4051 


4475 


3776 


49 


12 


5011 


1622 


6906 


8-2360908 


3670 


5234 


5639 


4921 


48 


13 


6299 


2888 


8150 


2130 


4872 


6417 


6803 


6067 


47 


14 


7587 


4153 


9393 


3353 


6075 


7600 


7966 


7212 


46 


15 


8874 


5418 


8-2290636 


4575 


7276 


8782 


9129 


8357 


45 


16 


8-2140161 


6682 


1879 


5796 


8478 


9964 


3-25S0292 


9501 


44 


17 


1447 


7946 


3121 


7018 


9679 


S-2511145 


1455 


8-2650645 


43 


18 


2733 


9210 


4363 


8239 8-2440880 


2326 


2617 


1789 


42 


19 


4019 


S-2220473 


5605 


9460 


2080 


3507 


3779 


2933 


41 


20 


5304 


1736 


6846 


8-2370680 


3280 


4688 


4941 


4076 


40 


21 


6589 


2998 


8087 


1900 


4480 


5868 


6102 


5219 


39 


22 


7874 


4260 


9327 


3120 


5680 


7048 


7263 


6362 


38 


23 


9158 


5522 


8-2300568 


4339 


6879 


8227 


8424 


7504 


37 


24 


8-2150442 


6784 


1807 


5558 


8077 


9407 


9584 


8646 


36 


25 


1725 


8045 


3047 


6776 


9276 


8-2520586 


8-2590744 


9788 


35 


26 


3008 


9305 


4286 


7995 


8-2450474 


1764 


1904 


8-2660929 


34 


27 


4291 


8-2230566 


5525 


9213 


1672 


2943 


3063 


2071 


33 


28 


5573 


1826 


6763 


8-2380430 


2869 


4121 


4223 


3212 


32 


29 


6855 


3085 


8001 


1648 


4066 


5298 


5381 


4352 


31 


30 


8137 


4345 


9239 


2865 


5263 


6476 


6540 


5492 


30 


31 


9418 


5604 


8-2310476 


4081 


6460 


7653 


7698 


6632 


29 


32 


3-2160699 


6862 


1713 


5297 


7656 


8829 


8856 


7772 


28 


33 


1979 


8120 


2950 


6513 


8852 


8-2530006 


8-2600014 


8911 


27 


34 


3259 


9378 


4186 


7729 


8-2460047 


1182 


1171 


8-267C051 


26 


35 


4539 


3-2240635 


5422 


8944 


1242 


2358 


2328 


1189 


25^ 


36 


5818 


1892 


6658 


8-2390159 


2437 


3533 


3485 


2328 


24 


37 


7097 


3149 


7893 


1373 


3632 


4708 


4641 


3466 


23 


38 


8375 


4405 


9128 


2588 


4826 


5883 


5797 


4604 


22 


39 


9653 


5661 


8-2320363 


3802 


6020 


7058 


6953 


5742 


21 


40 


B-2170931 


6917 


1597 


5015 


7213 


8232 


8108 


6879 


20 


41 


2209 


8172 


2831 


6228 


8407 


9406 


9263 


8016 


19 


42 


3486 


9427 


4064 


7441 


9599 


8-2540579 


8-2610418 


9153 


18^ 


43 


4762 


3-2250682 


5297 


8654 


3-2470792 


1752 


1573 


8-26S0289 


17 


44 


6038 


1936 


6530 


9866 


1984 


2925 


2727 


1425 


16 


45 


7314 


3190 


7763 


8-2401078 


3176 


4098 


3881 


2561 


15 


46 


8590 


4443 


8995 


2289 


4368 


5270 


5034 


3696 


14 


47 


9865 


5696 


8-2330227 


3500 


5559 


6442 


6188 


4832 


13 


48 


8-2181140 


6949 


1458 


4711 


6750 


7614 


7341 


5967 


12 


49 


2414 


8201 


2689 


5922 


7940 


8785 


8493 


7101 


11 


50 


3688 


9453 


3920 


7132 


9131 


9956 


9646 


8236 


10 


51 


4962 


3-2260705 


5150 


8342 


8-248032] 


8-2551127 


8-2620798 


9370 


9 


52 


6235 


1956 


6380 


9551 


1510 


2297 


1950 


8-2690503 


8 


53 


7508 


3207 


7610 


8-2410760 


2699 


3467 


3101 


1637 


7 


54 


8780 


4457 


8839 


1969 


3888 


4637 


4252 


2770 


6 


55 


8-2190053 


5708 


8-2340068 


3177 


5077 


5806 


5403 


3903 


5 


56 


1324 


6957 


1297 


4386 


6265 


6976 


6554 


5035 


4 


57 


2596 


8207 


2525 


5593 


7453 


8144 


7704 


6168 


3 


58 


3867 


9456 


3753 


6801 


8641 


9313 


8854 


7300 


2 


59 


5137 


8-2270705 


4980 


8008 


9828 


3-2560481 


3-2630004 


8431 


1 


60 


6408 


1953 


6208 


9215 


3-2491015 


1649 


1153 


9563 


A 


" 


3' 


2' 


V 


0' 


59' 


58' 1 57' I 


56' 




LOG. COTAN. 89°. LOG. COTAN. 88''. 


II 



^ 






LOG. SINE. 1°. 




\Tah 1. 




// 


4' 


W 


6' 


7' 


8' 


9' 


10' 


11' 1" 







8'2698810 


3-2766136 


8 -2332434 


8-2397734 


8-2962067 


8-3025460 


8-30S7941 


8-3149536 


60 




1 


9941 


7249 


3530 


8814 


3131 


6509 


8975 


8-3150555 


59 




2 


3-2701071 


8362 


4626 


9894 


4195 


7558 


8-3090009 


1574 


58 




3 


2201 


9475 


5722 


8-2900974 


5259 


8606 


1042 


2593 


57 




4 


3331 


8-2770587 


6818 


2053 


6322 


9654 


2075 


3611 


56 




5 


4461 


1700 


7913 


3132 


7385 


8-3030702 


3108 


4630 


55 




6 


5590 


2811 


9008 


4211 


8448 


1749 


4140 


5648 


54 




7 


6719 


3923 


8-2S40103 


5289 


9511 


2796 


5173 


6665 


53 




8 


7847 


5034 


1197 


6367 


3-2970573 


3843 


6205 


7683 


52 




9 


8976 


6145 


2292 


7445 


1635 


4890 


7237 


8700 


51 




10 


8-2710104 


7256 


3386 


8523 


2697 


5937 


6268 


9717 


50 




11 


1232 


8367 


4479 


9600 


3759 


6983 


9299 


8-3160734 


49 




12 


2359 


9477 


5573 


8-2910677 


4820 


8029 


8-3100330 


1751 


48 




13 


3486 


8-2780587 


6666 


1754 


5881 


9075 


1361 


2767 


47 




114 


4613 


1696 


7759 


2831 


6942 


8-3040120 


2392 


3783 


46 




15 


5740 


2806 


8851 


3907 


8002 


1165 


3422 


4799 


45 




16 


6866 


3915 


9943 


4983 


9063 


2210 


4452 


5815 


44 




17 


7992 


5023 


3-2851035 


6059 


3-2980123 


3255 


5482 


6830 


43 




18 


9118 


6132 


2127 


7134 


1183 


4299 


6512 


7845 


42 




19 


S-2720243 


7240 


3219 


8210 


2242 


5344 


7541 


8860 


41 




20 


1368 


8348 


4310 


9285 


3301 


6388 


8570 


9875 


40 




21 


2493 


9456 


5401 


8-2920359 


4360 


7431 


9599 


8-3170889 


39 




22 


3618 


8-2790563 


6491 


,1434 


5419 


8475 


8-3110628 


1903 


36 




23 


4742 


1670 


7582 


2508 


6477 


9518 


1656 


2917 


37 




24 


5866 


2777 


8672 


3582 


7536 


8-3050561 


2684 


3931 


36 




25 


6990 


3S83 


9762 


4656 


8594 


1604 


3712 


4945 


35 




26 


8113 


4989 


8-2860851 


5729 


9651 


2646 


4740 


5958 


34 




27 


9236 


6095 


1941 


6802 


8-2990709 


3688 


5767 


6971 


33 




28 


3-2730359 


7201 


3030 


7875 


1766 


4730 


6794 


7984 


32 




29 


1481 


8306 


4118 


6948 


2823 


5772 


7821 


8996 


31 




30 


2604 


9411 


5207 


8-2930020 


3879 


6813 


6848 


8-3180008 


30 




31 


3725 


8-2800516 


6295 


1092 


4936 


7855 


9874 


1021 


29 




32 


4847 


1621 


7383 


2164 


5992 


6896 


8-3120901 


2032 


28 




33 


5968 


2725 


8471 


3235 


7048 


9936 


1927 


3044 


27 




34 


7089 


3829 


9558 


4306 


8104 


8-3060977 


2952 


4055 


26 




35 


8210 


4933 


8-2870645 


5378 


9159 


2017 


3978 


5067 


25 




36 


9331 


6036 


1732 


6448 


8-3000214 


3057 


5003 


6077 


24 




37 


8-2740451 


7139 


2818 


7519 


1269 


4097 


6028 


7088 


23 




38 


1571 


8242 


3905 


8589 


2324 


5136 


7053 


8098 


22 




39 


2690 


9345 


4991 


9659 


3378 


6175 


8077 


9109 


21 




40 


3810 


8-2810447 


6076 


8-2940729 


4432 


7214 


9101 


8-3190119 


20 




41 


4929 


1549 


7162 


1798 


5486 


8253 


8-3130125 


1128 


19 




42 


6048 


2650 


8247 


2867 


6539 


9291 


1149 


2138 


18 




43 


7166 


3752 


9332 


3936 


7593 


8-3070330 


2173 


3147 


17 




44 


82S4 


4853 


8-28S0417 


5005 


8646 


1368 


3196 


4156 


16 




45 


9402 


5954 


1501 


6073 


9699 


2405 


4219 


5165 


15 




46 


S-2750520 


7055 


2585 


7141 


8-3O10751 


3443 


5242 


6173 


14 




47 


1637 


8155 


3669 


8209 


1804 


4480 


6264 


7182 


13 




48 


2754 


9255 


4752 


9277 


2856 


5517 


7287 


8190 


12 




!49 


3871 


8-2820355 


5836 


8-2950344 


3907 


6554 


8309 


9198 


11 




150 


4987 


' 1454 


6919 


1411 


4959 


7590 


9331 


8-3200205 


10 




51 


6103 


2553 


8002 


2478 


6010 


8626 


8-3140352 


1213 


9 




52 


7219 


3652 


9084 


3544 


7061 


9662 


1374 


2220 


8 




53 


8335 


4751 


S-2890166 


4611 


8112 


3-3080698 


2395 


3227 


7 




54 


9450 


5849 


1248 


5677 


9163 


1734 


3416 


4233 


6 




55 


8-2760565 


6947 


2330 


6742 


8-3020213 


2769 


4436 


5240 


5 




56 


1680 


8045 


3411 


7808 


1263 


3804 


5457 


6246 


4 




57 


2794 


9143 


4492 


8873 


2313 


4839 


6477 


7252 


3 




58 


3909 


S-2830240 


5573 


9938 


3362 


5873 


7497 


8258 


2 




59 


5022 


1337 


6654 8-2961003 


4411 


6907 


8516 


9263 


1 




60 


6136 


2434 


77341 2067 


5460 


7941 


9536 


8-3210269 







" 


55' 


54' J 


53' 1 52' 


51' 


50' 


49' 


48' 


// 







— — — 




LOG. COSINE 88°. 




1 





Table II.] 


LOG. TAN. 1°. 




si| 


4' 


5' 


6' 


7' 


8' 


9' 


10' 


11' 


// 


S-2699563 


8-2766912 


8-2833234 


8-2898559 


8-2962917 


8-3026335 


8-3088842 


8-3150462 


60 


1 8-2700694 


8026 


4331 


9640 


3981 


7385 


9876 


1482 


59 


2 1825 


9139 


5428 


8-2900720 


5046 


8433 


8-3090910 


2501 


58 


3 2955 


3-2770253 


6524 


1800 


611C 


9482 


1944 


3520 


57 


4 4085 


1365 


7620 


2879 


7174 


8-3030531 


2977 


4539 


56 


5 5215 


2478 


8716 


3959 


8237 


1579 


4010 


5558 


55 


6 6345 


3590 


9811 


5038 


9300 


2627 


5043 


6576 


54 


7 7474 


4702 


8-2S40906 


6117 


8-2970363 


3674 


6076 


7595 


53 


S 8603 


5814 


2001 


7195 


1426 


4722 


7109 


8613 


52 


9 9732 


6925 


3096 


8274 


2489 


576S 


8141 


9630 


51 


10 8-2710860 


8036 


4190 


9352 


3551 


6816 


9173 


8-3160648 


50 


11 1989 


9147 


5284 


8-2910430 


4613 


7862 


8-3100205 


1665 


49 


12 3116 


3-2780258 


6378 


1507 


5675 


8909 


1236 


2682 


48 


13 4244 


1368 


7471 


25f-4 


6736 


9955 


2267 


3699 


47 


14 5371 


2478 


8565 


3661 


7797 


8-3041001 


3298 


4715 


46 


15 6498 


3588 


9658 


4738 


885G 


2046 


4329 


5732 


45 


16 7625 


4697 


8-2850750 


5815 


9919 


3092 


5360 


6748 


44 


17 8751 


5806 


1843 


6891 


8-2980980 


4137 


6390 


7764 


43 


18 9877 


6915 


2935 


7967 


2040 


5182 


7420 


8779 


42 


19 8-2721003 


8024 


4027 


9042 


3100 


6226 


6450 


9795 


41 


20 2129 


9132 


5118 


8-2920118 


4159 


7271 


9479 


8-3170810 


40 


21 3254 


8-2790240 


6210 


1193 


5219 


8315 


8-3110508 


1825 


39 


22 4379 


1348 


7301 


2268 


6278 


9359 


1538 


2839 


38 


23 5504 


2455 


8392 


3342 


7337 


8-3050403 


2566 


3854 


37 


24 6628 


3563 


9482 


4417 


8395 


1446 


3595 


4868 


36 


25 7752 


4670 


8-2S60572 


5491 


9454 


2489 


4623 


5882 


35 


26 8876 


5776 


1662 


6565 


8-2990512 


3532 


5651 


6895 


34 


27 9999 


6882 


2752 


7638 


1570 


4575 


6679 


7909 


33 


28 3-2731122 


7988 


3841 


8711 


2627 


5617 


7707 


8922 


32 


29 2245 


9094 


4931 


9784 


3685 


6659 


8734 


9935 


31 


30 3368 


8-2800200 


6019 


8-2930857 


4742 


7701 


9761 


8-3180948 


30 


31 4490 


1305 


7108 


1930 


5799 


8743 


8-3120788 


1960 


29 


32 5612 


2410 


8196 


3002 


6855 


9784 


1815 


2973 


28 


33 6734 


3515 


9284 


4074 


7911 


8-3060825 


2841 


3985 


27 


34 7856 


4619 


8-2870372 


5145 


8967 


1866 


3867 


4997 


26 


35 8977 


5723 


1460 


6217 


8-3000023 


2907 


4893 


6008 


25 


36 8-2740098 


6827 


2547 


7288 


1079 


3947 


5919 


7019 


24 


37 1218 


7930 


3634 


8359 


2134 


4987 


6944 


8031 


23 


38 2338 


9034 


4720 


9429 


3189 


6027 


7969 


9041 


22 


39 3458 


8-2810136 


5807 


8-2940500 


4244 


7067 


8994 


8-3190052 


21 


40 4578 


1239 


6893 


1570 


5298 


8106 


8-3130019 


1062 


20 


41 5698 


2342 


7979 


2640 


6353 


9145 


1043 


2073 


19 


42 6817 


3444 


9065 


3709 


7407 


8-3070184 


2068 


3083 


18 


43 7936 


4545 


8-2880150 


4779 


8460 


1223 


3092 


4092 


17 


44 9054 


6647 


1235 


5848 


9514 


2261 


4115 


5102 


16 


45 8-2750173 


6748 


2320 


6916 


8-3010567 


3299 


5139 


6111 


15 


46 1291 


7849 


3404 


7985 


1620 


4337 


6162 


7120 


14 


47 2408 


8950 


4488 


9053 


2673 


5375 


7185 


8129 


13 


48 3526 


8-2820051 


5572 


8-2950121 


3725 


6412 


8208 


9137 


12 


49 4643 


1151 


6656 


1189 


4778 


7449 


9230 


8-3200145 


11 


50 5760 


2251 


7740 


2256 


5830 


8486 


8-3140253 


1154 


10 


51 6876 


3350 


8823 


3324 


6881 


9523 


1275 


2161 


9 


52 7992 


4450 


9906 


4391 


7933 


8-3080559 


2296 


3169 


8 


53 9108 


5549 


8-2890988 


5457 


8984 


1596 


3318 


4176 


7 


54 8-2760224 


6647 


2071 


6524 


8-3020035 


2631 


4339 


5183 


6 


55 1340 


7746 


3156 


7590 


1086 


3667 


5360 


6190 


5 


56 2455 


8844 


4235 


8656 


2136 


4703 


6381 


7197 


4 


57 3570 


9942 


5316 


9721 


3186 


5738 


7402 


8203 


3 


58 4684 


8 2831040 


6397 


3-2960787 


4236 


6773 


8422 


9210 


2 


59 5798 


2137 


7478 


1852 


5286 


7807 


9442 


8-3210215 


Ml 


•30 6912 


3234 


8559 


2917 


6335 


8842 


8 3150462 


1221 


H 


" 55' 54' 1 


53' 52' 1 


51' 


50' 


49' 


48' 


"W 




LOG. COTAN. 88<^ 




1 



82 




LOG. SINE. 1°. 






ITahle u.j 


// 


12' 


13' 


14' 


15' 


16' 


17' 


18' 


19' 


" 





^•3210269 


B-3270163 


8-3329243 


S-3387529 


8-3445043 


3-3501805 


3-3557835 


8-3613150 


60 


1 


1274 


1155 


3-3330221 


8494 


5995 


2745 


8762 


4066 


59 


2 


2278 


2146 


1199 


9459 


6947 


3685 


9690 


4982 


58 


3 


3283 


3137 


2176 


3-3390423 


7899 


4624 


8-3560617 


5897 


57 


4 


4287 


4127 


3153 


1387 


8351 


5563 


1544 


6813 


56 


5 


5292 


5118 


4130 


2351 


9802 


6502 


2471 


7728 


55 i 


6 


6295 


6108 


5107 


3315 


8-3450753 


7441 


3398 


8643 


54 


7 


7299 


7098 


6084 


4279 


1704 


8379 


4324 


9558 


531 


8 


8303 


8087 


7060 


5242 


2655 


9318 


5251 


8-3620472 


521 


9 


9306 


9077 


8036 


6205 


3605 


3-3510256 


6177 


1387 


5]! 


10 


8-3220309 


3-3280066 


9012 


7168 


4555 


1194 


7103 


2301 


50, 


11 


1311 


1055 


9988 


8131 


5505 


2132 


8029 


3215 


49 


12 


2314 


2044 


8-3340963 


9093 


6455 


3069 


8954 


4129 


48 i 


13 


3316 


3032 


1938 


8-3400055 


7405 


4006 


9880 


5042 


47 1 


14 


4318 


4021 


2913 


1018 


8354 


4944 


8-3570805 


5956 


46 r 


15 


5320 


5005 


3888 


1979 


9304 


5831 


1730 


6869 


45 


16 


6322 


5997 


4863 


2941 


8-3460253 


6817 


2654 


7782 


44 


17 


7323 


6984 


5837 


3902 


1201 


7754 


3579 


8695 


43 


18 


8324 


7972 


6811 


4864 


2150 


8690 


4503 


9603 


42 


19 


9325 


8959 


7785 


5825 


3098 


9626 


5427 


8-3630520 


41 


20 


8-3230326 


9946 


8759 


6785 


4047 


3-3520562 


6351 


1433 


40 


21 


1326 


8-3290933 


9732 


7746 


4995 


1498 


7275 


2345 


39 


22 


2326 


1919 


8-3350706 


8706 


5942 


2433 


8199 


3257 


38 


23 


3326 


2906 


1679 


9666 


6890 


3369 


9122 


4169 


37 


24 


4326 


3892 


2651 


8-3410626 


7837 


4304 


8-3580045 


5080 


36 


25 


5325 


4878 


3624 


1586 


8784 


5239 


0968 


5991 


35 


26 


6325 


5863 


4597 


2546 


9731 


6173 


1891 


6903 


34 


27 


7324 


6849 


5569 


. 3505 


8-3470678 


7108 


2814 


7814 


33 


28 


8322 


7834 


6541 


4464 


1625 


8042 


3736 


8724 


32 


29 


9321 


8819 


7512 


5423 


2571 


8976 


4658 


9635 


31 


30 


8-32i0319 


9804 


8484 


6382 


3517 


9910 


5580 


8-3640545 


30 


31 


1317 


8-3300788 


9455 


7340 


4463 


8-3530844 


6502 


1456 


29 


32 


2315 


1773 


8-3360426 


8298 


5409 


1778 


7424 


2366 


28 


33 


3313 


2757 


1397 


9256 


6354 


2711 


8345 


3275 


27 


34 


4310 


3740 


2368 


8-3420214 


7300 


3644 


9266 


4185 


26 


35 


5308 


4724 


3338 


1172 


8245 


4577 


8-3590187 


5095 


25 


36 


6305 


5708 


4309 


2129 


9189 


5510 


1108 


6004 


24 


37 


7301 


6691 


5279 


3086 


8-3480134 


6442 


2029 


6913 


23 


38 


8298 


7674 


6248 


4043 


1079 


7374 


2949 


7822 


22 


39 


9294 


8656 


7218 


5000 


2023 


8306 


3870 


8730 


21 


40 


8-3250290 


9639 


8187 


5957 


2967 


9238 


4790 


9639 


20 


41 


1286 


8-3310621 


9156 


6913 


3911 


8-3540170 


5709 


8-3650547 


19 


42 


2282 


1603 


8-3370125 


7869 


4854 


1102 


6629 


1455 


18 


43 


3277 


2585 


1094 


8825 


5798 


2033 


7549 


2363 


17 


44 


4272 


3567 


2063 


9781 


6741 


2064 


8468 


3271 


16 


45 


5267 


4548 


3031 


8-3430736 


7684 


3895 


9387 


4179 


15 


46 


6262 


5529 


3999 


1691 


8627 


4826 


8-3600306 


5086 


14 


47 


7256 


6510 


4967 


2646 


9570 


5756 


1225 


5993 


13 


48 


8250 


7491 


5934 


3601 


S-3490512 


6686 


2143 


6900 


12 


49 


9244 


8472 


6902 


4556 


1454 


7617 


3061 


7807 


11 


50 


8-3260238 


9452 


7869 


5510 


2396 


8546 


3979 


8713 


10 


51 


1232 


8-3320432 


8836 


6465 


3338 


9476 


4897 


9620 


9 


52 


2225 


1412 


9803 


7419 


4280 


8-3550406 


5815 


8-3660526 


8 


53 


3218 


2392 


8-3380769 


8372 


5221 


1335 


6733 


1432 


7 


54 


4211 


3371 


1736 


9326 


6162 


2264 


7650 


2338 


6 


55 


5204 


4350 


2702 


8-3440279 


7103 


3193 


8567 


3244 


5 


56 


6196 


5329 


3668 


1233 


8044 


4122 


9484 


4149 


4 


57 


7188 


6308 


4633 


2186 


8985 


5050 


8-3610401 


5054 


3 


58 


8180 


7287 


5599 


3138 


9925 


5979 


1317 


5959 


2 


59 


9172 


8265 


6564 


4091 


8-3500865 


6907 


2234 


6864 


1 


60 


8-3270163 


9243 


7529 


5043 


1805 


7835 


3150 


7769 





" 


47/ 


46' 


45' 


44' 


43' 


42' 


41' 


40' 




_ 






LOG, COSINE. 880 






. J| 



Table n.] 




LOG. TAN. 1°. 






sail 


" 


12' 


13' 


14' 


15' 


16' 


17' 


18' 


19' 


" 





S-3211221 


S-3271143 


3-3330249 


S-3388563 


8-3446105 


3-3502895 


8-3558953 


8-3614297 


60 


1 


2227 


2134 


1228 


9528 


7075 


3835 


9881 


5213 


59 


2 


3232 


3126 


2206 


8-3390493 


8010 


4775 


8-3560809 


6129 


58 


3 


4237 


4117 


3184 


1458 


8962 


5715 


1737 


7045 


57 


4 


5242 


5108 


4161 


2423 


9914 


6655 


2664 


7961 


56 


5 


6246 


6099 


5139 


3387 


8-3450866 


7594 


3592 


8877 


55 


6 


7251 


7090 


6116 


4351 


1817 


8533 


4519 


9793 


54 


7 


8255 


8080 


7093 


5316 


2769 


9472 


5446 


8-3620708 


53 


8 


9259 


9070 


8070 


6279 


3720 


8-3510411 


6373 


1623 


52 


9 


3-3220262 


s-3280060 


9046 


7243 


4671 


1350 


7299 


2538 


51 


10 


1266 


1050 


3-3340023 


8206 


5621 


2288 


8226 


3453 


50 


11 


2269 


2039 


0999 


9169 


6572 


3226 


9152 


4367 


49 


12 


3272 


3028 


1975 


8-3400132 


7522 


4164 


3-3570078 


B281 


48 


13 


4274 


4017 


2950 


1095 


8472 


5102 


1004 


6196 


47 


14 


5277 


5006 


3926 


2058 


9422 


6040 


1929 


7110 


46 


15 


6279 


5995 


4901 


3020 


8-3460372 


6977 


2855 


8023 


45 


16 


7231 


6983 


5876 


3982 


1321 


7914 


3780 


8937 


44 


17 


8283 


7971 


6851 


4944 


2271 


8851 


4705 


9850 


43 


19 


9285 


8959 


7826 


5906 


3220 


9788 


5630 


8-5630763 


42 


19 


3-3230286 


9947 


8800 


6867 


4169 


8-3520725 


6555 


1676 


41 


20 


1287 


3-3290934 


9774 


7828 


5117 


1661 


7479 


2589 


40 


21 


2288 


1921 


8-3350748 


8789 


€066 


2597 


8403 


3502 


39 


22 


3288 


2908 


1722 


9750 


7014 


3533 


9327 


4414 


38 


23 


4289 


3895 


2695 


3-3410711 


7962 


4469 


8-8580251 


5327 


37 


24 


5289 


4882 


3669 


1671 


8910 


5405 


1175 


6239 


36 


25 


6289 


5868 


4642 


2631 


9857 


6340 


2098 


7150 


35 


26 


7289 


6854 


5615 


3591 


8-3470805 


7275 


3022 


8062 


34 


27 


8288 


7840 


6587 


4551 


1752 


8210 


3945 


8974 


33 


28 


9287 


8826 


7560 


5511 


2699 


9145 


4868 


9885 


32 


29 


3-3»l0286 


9811 


8532 


6470 


3646 


8-3530080 


5790 


8-3640796 


31 


30 


1285 


S-3300796 


9504 


7429 


4592 


1014 


6713 


1707 


30 


31 


2284 


1781 


8-3360476 


8388 


5539 


1948 


7635 


2617 


29 


32 


3282 


2766 


1447 


9347 


6485 


2882 


8557 


3528 


28 


33 


4280 


3751 


2419 


8-3420305 


7431 


3816 


9479 


4438 


27 


34 


5278 


4735 


3390 


1263 


8377 


4750 


8-3590401 


5348 


26 


35 


6276 


5719 


4361 


2221 


9322 


5683 


1322 


6258 


25 


36 


7273 


6703 


5331 


3179 


8-3480268 


6616 


2243 


7168 


24 


37 


8270 


7687 


6302 


4137 


1213 


7549 


3165 


8078 


23 


38 


9267 


8670 


7272 


5094 


2158 


8482 


4086 


8987 


22 


39 


3-3250264 


9653 


8242 


6052 


3103 


9414 


5006 


9896 


21 


40 


1260 


8-3310636 


9212 


7009 


4047 


8-3540347 


5927 


8-3650805 


20 


41 


2257 


1619 


8-3370181 


7965 


4991 


1279 


6847 


1714 


19 


42 


3253 


2601 


1151 


8922 


5936 


2211 


7767 


2623 


18 


43 


4249 


3584 


2120 


9876 


6879 


3143 


8687 


3531 


17 


44 


5244 


4566 


3089 


8-3430835 


7823 


4074 


9607 


4439 


16 


45 


6240 


5548 


4058 


1791 


8767 


5006 


8.3600527 


5347 


15 


46 


7235 


6529 


5026 


2746 


9710 


5937 


1446 


6255 


14 


47 


8230 


7511 


5994 


3702 


8-3490653 


6868 


2365 


7163 


13 


48 


9224 


8492 


6963 


. 4657 


1596 


7799 


3284 


8070 


12 


49 


8-3260219 


9473 


7930 


5612 


2539 


8729 


4203 


8978 


11 


50 


1213 


8-3320454 


8898 


6567 


3481 


9660 


5121 


9885 


10 


51 


2207 


1434 


9866 


7522 


4423 


8-3550590 


6040 


8-3660792 


9 


5^ 


3201 


2415 


8-3380833 


8476 


5365 


1520 


6958 


1698 


8 


53 


4194 


3395 


1800 


9431 


6307 


2450 


7876 


2605 


7 


54 


5188 


4375 


2767 


8-3440385 


7249 


3379 


8794 


3511 


6 


55 


6181 


5354 


3733 


1339 


8191 


4309 


9711 


4417 


5 


56 


7173 


6334 


4700 


2292 


9132 


5238 


8-3610629 


5323 


4 


57 


8166 


7313 


5666 


3246 


8-3500073 


6167 


1546 


6229 


3 


58 


9158 


8292 


6632 


4199 


1014 


7096 


2463 


7135 


2 


59 


8-3270151 


9271 


7597 


5152 


1954 


8024 


3380 


8040 


1 


60 


1143 


8-3330249 


8563 


6105 


2895 


8953 


4297 


8945 Oil 




4r 


46' 


45' 


44' 1 


43' 


43' 


41' 


40' " 1 


I,. - 




LOG. COTAN. 88°. 




-a— __ 


— ^J 



84 






LOG. SINE 1°. 




[Table 


[I. 


// 


20' 


21' 


22' 


23' 


24' 


25' 


26' 27' 


T, 





8-3667769 


8-3721710 


8-3774988 


8-3827620 


8-3879622 


8-3931008 


8-3981793 


B-4031990 


60 


1 


8674 


2603 


5870 


8492 


8-3880483 


1859 


2634 


2822 


59 


2 


9578 


3496 


6753 


9364 


1345 


2710 


3475 


3653 


58 


3 


8-3670482 


4389 


7635 


8-3830235 


2206 


3561 


4316 


4485 


57 


I 4 


1386 


5282 


8517 


1106 


3067 


4412 


5157 


5316 


56 


5 


2290 


6174 


9398 


1978 


3927 


5263 


5998 


6147 


55 


6 


3193 


7067 


3-3780280 


2848 


4788 


6113 


6839 


6978 


54 


7 


4097 


7959 


1161 


3719 


5648 


6964 


7679 


7809 


53 


8 


5000 


8851 


2042 


4590 


6509 


7814 


8519 


8639 


52 


9 


5903 


6743 


2924 


5460 


7369 


8664 


9359 


9470 


51 


10 


6806 


8-3730635 


3804 


6330 


8229 


9513 


8-3990199 


8 4040300 


50 


11 


7708 


1526 


4685 


7201 


9088 


8-3940363 


1039 


1130 


49 


12 


8611 


2418 


5566 


8070 


9948 


1213 


1879 


1960 


48 


13 


9513 


3309 


6446 


8940 


8-3890807 


2062 


2718 


2790 


47 


14 


8-3680415 


4200 


7326 


9810 


1666 


2911 


3557 


3620 


46 


15 


1317 


5091 


8206 


8-3840679 


2526 


3760 


4397 


4449 


45 


16 


2219 


5981 


9086 


1548 


3384 


4609 


5236 


5279 


44 


17 


3120 


6872 


9965 


2417 


4243 


5457 


6074 


6108 


43 


18 


4022 


7762 


3-3790845 


3286 


5102 


6306 


6913 


6937 


42 


19 


4923 


8652 


1724 


4155 


5960 


7154 


7751 


7766 


41 


20 


5824 


9542 


2603 


5023 


6818 


8002 


8590 


8594 


40 


21 


6725 


8-3740431 


3482 


5892 


7676 


8850 


9428 


9423 


39 


22 


7625 


1321 


4361 


6760 


8534 


9698 


8-4000266 


8-4050251 


38 


23 


8526 


2210 


5239 


7628 


9392 


8-3950546 


1104 


1080 


37 


24 


9426 


3099 


6117 


8496 


8-3900249 


1393 


1941 


1908 


36 


25 


8-3690326 


3988 


6996 


9363 


1107 


2240 


2779 


2736 


35 


26 


1226 


4877 


7874 


8-3850231 


1964 


3088 


3616 


3563 


34 


27 


2125 


5766 


8751 


1098 


2821 


3935 


4453 


4391 


33 


28 


3025 


6654 


9629 


1965 


3678 


4781 


5290 


5218 


32 


29 


3924 


7542 


8-3800507 


2832 


4534 


5628 


6127 


6046 


31 


30 


4823 


8430 


1384 


3699 


5391 


6475 


6964 


6873 


30 


31 


5722 


9318 


2261 


4565 


6247 


7321 


7801 


7700 


29 


32 


6621 


8-3750206 


3138 


5432 


7103 


8167 


8637 


8527 


28 


33 


7519 


1094 


4015 


6298 


7959 


9013 


9473 


9353 


27 


34 


8418 


1981 


4891 


7164 


8815 


9859 


8-4010309 


8-4O6O180 


26 


35 


9316 


2868 


5768 


8030 


9671 


8-3960705 


1145 


1006 


25 


36 


8-3700214 


3755 


6644 


8896 


8-3910526 


1550 


1981 


1832 


24 


37 


nil 


4642 


7520 


9761 


1382 


2395 


2816 


265? 


23 


38 


2009 


5528 


8396 


8-3860627 


2237 


3241 


3652 


3484 


22 


39 


2907 


6415 


9271 


1492 


3092 


4086 


4487 


431C 


21 


40 


3804 


7301 


8-3810147 


2357 


3947 


4930 


5322 


513£ 


20 


41 


4701 


8187 


1022 


3222 


4801 


5775 


6157 


5961 


19 


42 


5598 


9073 


1897 


4087 


5656 


6620 


6992 


6786 


18 


43 


6494 


9959 


2772 


4951 


6510 


7464 


7826 


761] 


17. 


44 


7391 


8-3760844 


3647 


5816 


7364 


8308 


8661 


843£ 


16" 


45 


8287 


1729 


4522 


6680 


8218 


9152 


9495 


9261 


15 


46 


9183 


2615 


5396 


7544 


9072 


9996 


8-4020329 


8-407008E 


14 


47 


8-3710079 


3500 


6271 


8408 


9926 


8-3970840 


1163 


091C 


13 


48 


0975 


4384 


7145 


9271 


a-3920779 


1683 


1997 


173^ 


12 


49 


1870 


5269 


8019 


83870135 


1633 


2527 


2831 


255£ 


Ml 


50 


2766 


6153 


8892 


0998 


2486 


3370 


3664 


3385 


no 


51 


3661 


7038 


9766 


1861 


3339 


4213 


4497 


420f 


9 


52 


4556 


7922 


8-3820639 


2724 


4191 


5056 


5331 


503C 


) 8 


53 


5451 


8806 


1513 


3587 


5044 


5898 


6164 


585- 


} 7 


54 


6346 


9689 


2386 


4450 


5897 


6741 


6996 


667: 


6 


55 


7240 


8-37r0573 


3258 


5312 


6749 


7583 


7829 


750C 


5 


56 


8134 


1456 


4131 


6174 


7601 


8425 


8662 


832S 


4 


57 


9028 


2339 


5004 


7037 


8453 


9268 


9494 


914e 


3 


58 


9922 


3222 


5876 


7898 9305 


8-3980109 


8-4030326 


996S 


2 


59 


8-3720816 


4105 


6745 


8760 8-3930156 


0951 


1158 


8-4080791 


1 


60 


1710 


4988 


7620 


9622 1008 


1793 


1990 


1614 





1 


39' 


38' 


37' 


36' 35' 


34' 


33' 


32' 




f- 






LOG. COSINE 88° 


• 


_JI 





Table 11/ 






LOG. TAN. 1°. 






85 1 




f 


20' 


21' 


22' 


23' 


24' 


25' 


26' 


27' " 




( 


3-3t56394f 


S-37229n 


) 8-377622: 


3-3828886 


8-38809ie 


8-3932336 


8-3983152 


8-4033381 60 




j 


935C 


380' 


) 7101 


9755 


178C 


3187 


3994 


4213 59 






3-367075J 


470: 


\ 798£ 


8-3830631 


2642 


403c 


4835 


5045 58 




'• 


166C 


5596 


) 8872 


1503 


3504 


4891 


5677 


5877 57 




A 


256-^ 


648f 


) 9754 


2374 


4365 


5742 


6519 


6709 56 




e 


346£ 


738: 


} 3-3780636 


3246 


5227 


6593 


7360 


7541 55 




6 


4372 


827f 


151£ 


4117 


6085 


7444 


8201 


8372 54 




7 


5276 


9i6e 


240C 


4989 


694E 


8295 


9042 


9203 53 




6 


618C 


3-373006] 


328S 


5860 


7809 


9145 


9883 


8-4040035 52 




9 


7082 


095S 


4164 


6731 


8670 


9996 


8-3990723 


0866 51 




10 


7987 


1845 


5045 


7601 


9530 


8-3940846 


1564 


1696 50 




11 


8890 


2737 


5926 


8472 


8-3890391 


1696 


2404 


2527 49 




12 


9793 


362S 


6807 


9342 


1251 


2546 


3244 


3358 48 




13 


8-3680696 


4521 


7688 


8-3840213 


2111 


3396 


4084 


4188 47 




14 


1598 


5412 


8569 


1083 


2970 


4246 


4924 


5018 46 




15 


2501 


6304 


9449 


1953 


3830 


5095 


5764 


5848 45 




16 


3403 


7195 


8-3790329 


2822 


4689 


5945 


6603 


6678 44 




17 


4305 


808G 


1209 


3692 


5548 


6794 


7442 


7508 43 




18 


5207 


8976 


2089 


4561 


6408 


7643 


8282 


8337 42 




19 


6108 


9867 


2969 


5430 


7266 


8492 


9121 


9167 41 




20 


7010 


8-3740757 


3849 


6299 


8125 


9340 


9959 


9996 40 




21 


7911 


1647 


4728 


7168 


8984 


8-3950189 


8-4000798 


8-4050825 39 




22 


8812 


2538 


5607 


8037 


9842 


1037 


1637 


1654 38 




23 


9713 


3427 


6486 


8905 


8-3900700 


1885 


2475 


2483 37 




24 


8-3690614 


4317 


7365 


9774 


1558 


2733 


3313 


3311 36 




25 


1514 


5206 


8244 


8-3850642 


2416 


3581 


4151 


4140 35 




26 


2414 


6096 


9122 


1510 


3274 


4429 


4989 


4968 34 




27 


3315 


6985 


8-3800001 


2378 


4131 


5276 


5827 


5796 33 




28 


4215 


7874 


0879 


3245 


4989 


6124 


6664 


6624 32 




29 


5114 


8762 


1757 


4113 


5846 


6971 


7502 


7452 31 




30 


6014 


9651 


2634 


4980 


6703 


7818 


8339 


8280 30 




31 


6913 


8-3750539 


3512 


5847 


7560 


8665 


9176 


9107 29 




32 


7812 


1428 


4390 


6714 


8417 


9511 


8-4010013 


9935 28 




33 


8711 


2316 


5267 


7581 


9273 


8-3960358 


0850 


8-4060762 27 




34 


9610 


3203 


6144 


8448 


8-3910129 


1204 


1686 


1589 26 




35 


8-3700509 


4091 


7021 


9314 


0986 


2050 


2523 


2416 25 




36 


1407 


4979 


7898 


8-3860180 


1842 


2897 


3359 


3242 24 




37 


2306 


5866 


8774 


1046 


2697 


3742 


4195 


4069 23 




38 


3204 


6753 


9650 


1912 


3553 


4588 


5031 


4895 22 




39 


4102 


7640 


8-3810527 


2778 


4409 


5434 


5867 


5722 21 




40 


4999 


8527 


1403 


3643 


5264 


6279 


6702 


6548 20 




41 


5897 


9413 


2278 


4509 


6119 


7124 


7538 


7374 19 




42 


6794 


8-3760299 


3154 


5374 


6974 


7969 


8373 


8199 18 




43 


7692 


1186 


4030 


6239 


7829 


8814 


9208 


9025 17 




44 


8589 


2072 


4905 


7104 


8684 


9659 


8-4020043 


9850 16 




45 


9485 


2958 


5780 


7969 


9538 


8-3970503 


0878 


8-4070676 15' 




46 


8-3710382 


3843 


6655 


8833 


8-3920393 


1348 


1713 


1501 14 




47 


1278 


4729 


7530 


9698 


1247 


2192 


2547 


2326 13 




48 


2175 


5614 


.8404 


S-3870562 


2101 


3036 


3381 


3151 12 




49 


3071 


6499 


9279 


1426 


2955 


3880 


4216 


3975 11 




50 


3967 


7384 


8-3820153 


2290 


3808 


4724 


5050 


4800 iO 




51 


4862 


8269 


1027 


3153 


4662 


5567 


5884 


5624 9 




52 


5758 


9153 


1901 


4017 


5515 


6411 


6717 


6449 8 




53 


6653 


3-3770038 


2775 


4880 


6368 


7254 


7551 


7273 7 




54 


7548 


0922 


3648 


5743 


7221 


8097 


8384 


8097 6' 




55 


8443 


1806 


4522 


6606 


8074 


8940 


9217 


8920 5 




56 


9338 


2690 


5395 


7469 


8927 


9782 6 


•4030050 


9744 4 




57 


3-3720232 


3574 


6268 


8332 


9779^ 


3-3980625 


0883 S 


-4080567 3 




58 


1127 


4457 


7141 


9194 


3-3930631 


1467 


1716 


1391 2 




59 


2021 


5340 


8014^ 


3-3880056 


1484 


2310 


2549 


2214 1 




60 


2915 


6223 


8886 


0918 


2336 


3152 


3381 


3037 1 




// 


39' 


38' 


37' 


36' 1 


35' 


34' 


33' 1 


32' " 1 






^_____. 


] 


LOG. COTAN. 88'='. 


_^___ 




Jl 



8* 





J 


S 






LOG. SINE lo. 






[Table u. 






28' 


29' 


30' 


31' 


32' 


33' 


34' 


35' 











8-4081614 


84130676 


8-4179190 


8-4227168 


8-4274621 


8-4321561 


8-4367999 


8-4413944 


60 






1 


2436 


1489 


9994 


7963 


5408 


2339 


8768 


4706 


59 






2 


3258 


2302 


8-4180798 


8758 


6194 


3117 


9538 


5468 


58 






3 


4080 


3115 


1602 


9553 


6980 


3895 


6-4370307 


6229 


57 






4 


4902 


3927 


2405 


8-4230348 


7766 


4672 


1077 


6990 


56 






5 


5723 


4740 


3209 


1142 


8552 


5450 


1846 


7751 


55 






6 


6545 


5552 


4012 


1937 


9338 


6227 


2615 


8512 


54 






7 


7366 


6364 


4815 


2731 


8-4280124 


7004 


3384 


9273 


53 






8 


8187 


7176 


5618 


3525 


0909 


7781 


4153 


8-4420034 


52 






9 


9008 


7988 


6421 


4319 


1694 


8558 


4921 


0795 


51 






10 


9829 


8800 


7223 


5113 


2480 


9335 


5690 


1555 


50 






11 


8-4090650 


9611 


8026 


5907 


3265 


8-4330112 


6458 


2315 


49 






12 


1471 


8-4140422 


8828 


6700 


4050 


0888 


7227 


3076 


48 






13 


2291 


1234 


9630 


7494 


4835 


1665 


. 7995 


3836 


47 






14 


3111 


2045 


84190432 


8287 


5619 


2441 


8763 


4596 


46 






15 


3931 


2856 


1234 


9080 


6404 


3217 


9531 


5855 


45 






16 


4751 


3666 


2036 


9873 


7188 


3993 


8-4380298 


6115 


44 






17 


5571 


4477 


2838 


8-4240666 


7972 


4769 


1066 


6875 


43 






18 


6391 


5287 


3639 


1458 


8756 


5544 


1833 


7634 


42 






19 


7210 


6098 


4441 


2251 


9540 


6220 


2601 


8393 


41 






20 


8029 


6908 


5242 


3043 


8-4290324 


7095 


3368 


9152 


40 






21 


8849 


7718 


6043 


3836 


1108 


7871 


4135 


9911 


39 






22 


9668 


8528 


6844 


4628 


1891 


8646 


4902 


8-4430670 


38 






23 


84100486 


9337 


7644 


5420 


2675 


9421 


5669 


1429 


37 






24 


1305 


8-4150147 


8445 


6211 


3458 


8-4340196 


6435 


2187 


36 






25 


2124 


0956 


9245 


7003 


4241 


0970 


7202 


2946 


35 






26 


2942 


1765 


8-4200046 


7795 


5024 


1745 


7968 


3704 


34 






27 


3760 


2575 


0846 


8586 


5807 


2519 


8734 


4462 


33 






28 


4578 


3383 


1646 


9377 


6590 


3294 


9501 


5221 


32 






29 


5396 


4192 


2440 


8-4250168 


7372 


4068 


8-4390266 


5978 


31 






30 


6214 


5001 


3245 


0959 


8154 


4842 


1032 


6736 


30 






31 


7032 


5809 


4045 


1750 


8937 


5616 


1798 


7494 


29 






32 


7849 


6618 


4844 


2541 


9719 


6389 


2564 


8251 


28 






33 


8657 


7426 


5644 


3331 


8-4300501 


7163 


3329 


9009 


27 






34 


9484 


8234 


6443 


4122 


1283 


7937 


4094 


9766 


26 






35 


84110301 


9042 


7242 


4912 


2064 


8710 


4859 


8-4440523 


25 






36 


1118 


9850 


8040 


5702 


2846 


9483 


5624 


1280 


24 






37 


1934 


84160657 


8839 


6492 


3627 


8-4350256 


6389 


2037 


23 






38 


2751 


1465 


9638 


7282 


4409 


1029 


7154 


2794 


22 






39 


3567 


2272 


8-4210436 


8071 


5190 


1802 


7919 


3551 


21 






40 


4383 


3079 


1234 


8861 


5971 


2574 


8683 


4307 


20 






41 


5200 


3886 


2032 


9650 


6751 


3347 


9447 


5063 


19 






42 


6015 


4693 


2830 


8-4260439 


7532 


4119 


8-4400212 


5820 


18 






43 


6831 


5499 


3628 


1229 


8313 


4892 


0976 


6576 


17 






44 


7647 


6306 


4426 


2018 


9093 


5664 


1740 


7332 


16 






45 


8462 


7112 


5223 


2806 


9873 


6436 


2503 


8087 


15 






4-6 


9278 


7919 


6020 


3595 


8-4310654 


7207 


3267 


8843 


14 






47 


84120093 


8725 


6818 


4383 


1434 


7979 


4031 


9599 


13 






48 


0908 


9531 


7615 


5172 


2213 


8751 


4794 


8-4450354 


12 






49 


1723 


8-4170336 


8412 


5960 


2993 


9522 


5557 


1109 


U 






50 


2537 


1142 


9208 


6748 


3773 


8-4360293 


6321 


1865 


10 






51 


3352 


1948 


8-4220005 


7536 


4552 


1064 


7083 


2620 


9 






52 


4166 


2753 


0801 


8324 


5332 


1835 


7846 


3375 


8 






53 


4981 


3558 


1598 


9111 


6111 


2606 


8609 


4129 


7 






54 


5795 


4363 


2394 


9899 


6890 


3377 


9372 


4884 


6 






55 


6609 


5168 


3190 


84270686 


7669 


4148 


84410134 


5638 


5 






56 


7422 


5973 


3986 


1474 


8447 


4918 


0896 


6393 


4 






57 


8236 


6777 


4782 


2261 


9226 


5688 


1659 


7147 


3 






58 


9050 


7582 


5577 


3048 


8-4320004 


6459 


2421 


7901 


2 






59 


9863 


8386 


6373 


3834 


0783 


7229 


3183 


8655 


1 






60 


84130676 


9190 


7168 


4621 


1561 


7999 


3944 


9409 









// 


3F 


30' 


29' 


28' 


27' 


26' 


25' 


24' 


" 






_ 






»»»a«M« 


LOG. COSINE 88< 


3, 


______ 









Table II.] 






LOG. TAN. 1°. 






87] 






~ 


28' 


29' 


30' 


31' 


32' 


33' 


34' 


35' 


// 









8 4083037 


8-4132132 


8-4180679 


S-4228690 


8-4276176 


8-4323150 


8-4369622 


8-4415603 


60 






1 


3859 


2945 


1483 


9485 


6963 


3929 


8-4370393 


6365 


59 






2 


4682 


3759 


2288 


8-4230281 


7750 


4707 


1163 


7127 


58 






3 


5505 


4572 


3092 


1076 


8537 


5486 


1933 


7889 


57 






4 


6327 


5385 


3896 


1872 


9324 


6264 


2703 


8651 


56 






5 


7149 


6198 


4700 


2667 


8-4280110 


7042 


3473 


9413 


55 






6 


7971 


7011 


5504 


3462 


0897 


7820 


4242 


8-4420174 


54 






7 


8793 


7823 


6307 


4257 


1683 


8598 


5012 


0936 


53 






8 


9615 


8636 


7111 


5051 


2469 


9375 


5781 


1697 


52 






9 


3-4090436 


9448 


7914 


5846 


3255 


8-4330153 


6550 


2458 


51 






10 


1258 


8-4140261 


8717 


6640 


4041 


0930 


7320 


3219 


50 






11 


2079 


1073 


9520 


7434 


4826 


1707 


8089 


3980 


49 






12 


2900 


1885 


8-4190323 


8229 


5612 


2484 


8857 


4741 


48 






13 


3721 


2696 


1126 


9023 


6397 


3261 


9626 


5502 


47 






14 


4542 


3508 


1929 


9816 


7182 


4038 


8-4380395 


6262 


46 






15 


5362 


4319 


2731 


8-4240610 


7968 


4815 


1163 


7023 


45 






16 


6183 


5131 


3533 


1404 


8752 


5591 


1931 


7783 


44 






17 


7003 


5942 


4336 


2197 


9537 


6368 


2700 


8543 


43 






18 


7823 


6753 


5138 


2990 


8-4290322 


7144 


3468 


9303 


42 






19 


8643 


7564 


5940 


3783 


1106 


7920 


4235 


8-4430063 


41 






20 


9463 


8374 


6741 


4576 


1891 


8696 


5003 


0822 


40 






21 


8-4100283 


9185 


7543 


5369 


2675 


9472 


5771 


1582 


39 






22 


1103 


9995 


8344 


6162 


3459 


8-4340248 


6538 


2341 


38 






23 


1922 


8-4150805 


9146 


6954 


4243 


1023 


7306 


3101 


37 






24 


2741 


1616 


9947 


7747 


5027 


1799 


8073 


3860 


36 






25 


3560 


2425 


8-4200748 


8539 


5811 


2574 


8840 


4619 


35 






26 


4379 


3235 


1549 


9331 


6694 


3349 


9607 


5378 


34 






27 


5198 


4045 


2349 


8-4250123 


7377 


4124 


8-4390374 


6137 


33 






28 


6017 


4854 


3150 


0915 


8161 


4899 


1140 


6895 


32 






29 


6835 


5664 


3950 


1706 


8944 


5674 


1907 


7654 


31 






30 


7653 


6473 


4750 


2498 


9727 


6448 


2673 


8412 


30 






31 


8472 


7282 


5550 


3289 


8-4300510 


7223 


3440 


9171 


29 






32 


9290 


8091 


6350 


4080 


1292 


7997 


4206 


9929 


28 






33 


8-4110107 


8900 


7150 


4872 


2075 


8771 


4972 


8-4440687 


27 






34 


0925 


9708 


7950 


5662 


2857 


9545 


5738 


1444 


26 






35 


1743 


8-4160517 


8749 


6453 


3639 


8-4350319 


6503 


2202 


25 






36 


2560 


1325 


9549 


7244 


4422 


1093 


7269 


2960 


24 






37 


3377 


2133 


8-4210348 


8034 


5204 


1867 


8034 


3717 


23 






38 


4194 


2941 


1147 


8825 


5985 


2640 


8800 


4475 


22 






39 


5011 


3749 


1946 


9615 


6767 


3413 


9565 


5232 


21 






40 


5828 


4556 


2745 


8-4260405 


7549 


4167 


8-4400330 


5989 


20 






41 


6645 


5364 


3543 


1195 


8330 


4960 


1095 


6746 


19 






42 


7461 


6171 


4342 


1985 


9111 


5733 


1860 


7503 


18 






43 


8278 


6979 


5140 


2774 


9892 


6506 


2624 


8259 


17 






44 


9094 


7786 


5938 


3564 


8-4310673 


7278 


3389 


9016 


16 






45 


9910 


8593 


6736 


4353 


1454 


8051 


4153 


9772 


15 






46 


8-4120726 


9399 


7534 


5142 


2235 


8823 


4918 


8-4450529 


14 






47 


1541 


8-4170206 


8332 


5932 


3016 


9595 


5682 


1285 


13 






48 


2357 


1012 


9130 


6720 


3796 


8-4360367 


6446 


2041 


12 






49 


3172 


1819 


9927 


7509 


4576 


1139 


7209 


2797 


11 






50 


3988 


2625 


8-4220725 


8298 


5356 


1911 


7973 


3552 


10 






51 


4803 


3431 


1522 


9086 


6136 


2683 


8737 


4308 


9 






52 


5618 


4237 


2319 


9875 


6916 


3455 


9500 


5063 


8 






53 


6432 


5043 


3116 


8-4270663 


7696 


4226 


8-4410263 


5819 


7 






54 


7247 


5848 


3912 


1451 


8476 


4997 


1027 


6574 


6 






55 


8062 


6654 


4709 


2239 


9255 


5768 


1790 


7329 


5 






56 


8876 


7459 


5505 


3027 


8-4320034 


6540 


2553 


8084 


4 






57 


9690 


8264 


6302 


3814 


0814 


7310 


3315 


8839 


3 






58 


8-4130504 


9069 


7098 


4602 


1593 


8081 


4078 


9594 


2 






59 


1318 


9874 


7894 


5389 


2372 


8852 


4841 


8-4460348 


1 






60 


2132 


8-4180679 


8690 


6176 


3150 


9622 


5603 


1103 









" 


31' 


30' 


29' 


28' 


27' 


26' 


25' 


24' 


" 












LOG. COTAN. 88® 











86 


1 







LOG. SINE 1°. 


_ 




{Table 


11. 




/' 


36' 


37' 


38' 


39-' 


40' 


41' 


42' 


43' 


" 







8-4459409 


8-4504402 


3-4548934 


3-4593013 


S-4636649 


8-4679850 


S-4722626 


8-4764984 


60 




1 


3-4460163 


5146 


9672 


3744 


7372 


8-4680567 


3335 


5686 


59 




2 


0916 


5894 


8-4550410 


4474 


8096 


1283 


4044 


6388 


58 




3 


1670 


6640 


1148 


5205 


8819 


1999 


4753 


7091 


57 




4 


2423 


7385 


1886 


5936 


9542 


2715 


5462 


7793 


56 




5 


3176 


8131 


2624 


6666 


8-4540265 


3431 


6171 


8495 


55 




6 


3929 


8876 


3362 


7396 


0988 


4147 


6860 


9197 


54 




7 


4682 


9621 


4099 


8126 


1711 


4862 


7589 


9899 


53 




8 


5435 


8-4510366 


4837 


8856 


2434 


5578 


8297 


8-4770600 


52 




9 


6188 


nil 


5574 


9586 


3156 


6293 


9006 


1302 


51 




10 


6940 


1856 


6311 


8-4600316 


3879 


7009 


9714 


2003 


50 




11 


7693 


2601 


7048 


1046 


4601 


7724 


8-4730422 


2705 


49 




12 


8445 


3345 


7785 


1775 


5323 


8439 


1130 


3406 


48 




13 


9197 


4090 


8522 


2505 


6046 


9154 


1838 


4107 


47 




14 


9949 


4834 


9259 


3234 


6768 


9869 


2546 


4808 


46 




15 


8-4470701 


5578 


9996 


3963 


7489 


3-4690584 


3254 


5509 


45 




16 


1453 


6322 


3-4560732 


4692 


8211 


1298 


3962 


6210 


44 




17 


2205 


7066 


1468 


5421 


8933 


2013 


4669 


6910 


43 




18 


2956 


7810 


2205 


6150 


9654 


2727 


5377 


7611 


42 




19 


3707 


8553 


2941 


6878 


8-4650376 


3441 


6084 


8311 


41 




20 


4459 


9297 


3677 


7607 


1097 


4156 


6791 


9012 


40 




21 


5210 


8-4520040 


4412 


8335 


1818 


4870 


7498 


9712 


39 




22 


5961 


0784 


5148 


9064 


2539 


5583 


8205 


8-4780412 


38 




23 


6712 


1527 


58S4 


9792 


3260 


6297 


8912 


1112 


37 




24 


7462 


2270 


6619 


3-4610520 


3981 


7011 


9618 


1812 


35 




25 


8213 


3013 


7354 


1248 


4702 


7725 


8-4740325 


2511 


35 




26 


8963 


3755 


8090 


1975 


5422 


8438 


1032 


3211 


34 




27 


9714 


4498 


8825 


2703 


6143 


9151 


1738 


3911 


33 




28 


8-4480464 


5240 


9560 


3431 


6863 


9865 


2444 


4610 


32 




29 


1214 


5983 


3-4570295 


4158 


7583 


8-4700576 


3150 


5309 


31 




30 


1964 


6725 


1029 


4886 


8303 


1291 


3856 


6009 


30 




31 


2714 


7467 


1764 


5613 


9023 


2003 


4562 


6708 


29 




32 


3463 


8209 


2498 


6340 


9743 


2716 


5268 


7407 


28 




33 


4213 


8951 


3233 


7067 


3-4660463 


3429 


5974 


8105 


27 




34 


4962 


9693 


3967 


7794 


1182 


4141 


6679 


8604 


26 




35 


5712 


8-4530434 


4701 


8520 


1902 


4854 


7385 


9503 


25 




36 


6461 


- 1176 


5435 


9247 


2621 


5566 


8090 


S-4790201 


24 




37 


7210 


1917 


6169 


9973 


3340 


6278 


8795 


0900 


23 




38 


7959 


2659 


6902 


3-4620700 


4059 


6990 


9500 


1598 


22 




39 


8708 


3400 


7636 


1426 


4778 


7702 


8-4750205 


2296 


21 




40 


9456 


4141 


8369 


2152 


5497 


8414 


0910 


2994 


20 




41 


8-4490205 


4881 


9103 


2878 


6216 


9126 


1615 


3692 


19 




42 


0953 


5622 


9836 


3604 


6935 


9837 


2320 


4390 


18 




43 


1701 


6363 


8-4580569 


4330 


7653 


8-4710549 


3024 


5088 


17 




44 


2450 


7103 


1302 


5055 


8372 


1260 


3729 


5785 


16 




45 


3198 


7844 


2035 


5781 


9090 


1971 


4433 


6483 


15 




46 


3945 


8584 


2768 


6506 


9808 


2682 


5137 


7180 


14 




47 


4693 


9324 


3500 


7231 


3-4670526 


3393 


5841 


7878 


13 




48 


5441 


8-4540064 


4233 


7957 


1244 


4104 


6545 


8575 


12 




49 


6188 


0804 


4965 


8682 


1962 


4815 


7249 


9272 


11 




50 


6936 


1543 


5697 


9406 


2680 


5526 


7953 


9969 


10 




51 


7683 


2283 


6429 


8-4630131 


3397 


6236 


8656 


8-4S00666 


9 




52 


8430 


3023 


7161 


0856 


4115 


6947 


9360 


1362 


8 




53 


9177 


3762 


7893 


1580 


4832 


7657 


8-4760063 


2059 


7 




[54 


9924 


4501 


8625 


2305 


5549 


8367 


0766 


2755 


6 




55 


8-4500671 


5240 


9357 


3029 


6266 


9077 


1470 


3452 


5 




56 


1417 


5979 


8-45900S8 


3753 


6983 


9787 


2173 


4148 


4 




57 


2164 


6718 


0819 


4477 


7700 


8-4720497 


2876 


4844 


3 




58 


2910 


7457 


1551 


5201 


8417 


1207 


3578 


5540 


2 




59 


3656 


8195 


2282 


5925 


9134 


1916 


4281 


6236 


I 




60 


4402 


6934 


3013 


6649 


9650 


2626 


4964 


6932 







" 


23' 


22' 


2r 


20' 


19'. 


18' 


17' 


16' 


' ' 













LOG. COS 


INE 8S°. 






' 





Table II.] 






LOG. TAN. 1°. 








89 


" 


36' 


r 37' 


38' 


39' 


40' 


41' 


42' 


43' 


// 





8-4461103 


8-4506131 


8-4550699 


8-4594814 


84638486 


8-4681725 


8-4724538 


8-4766933 


60 


] 


1857 


6878 


1438 


5545 


9211 


2442 


5248 


7636 


59 


1 2 


2611 


7624 


2176 


6277 


9935 


3159 


5957 


8339 


58 


3 


3365 


8371 


2915 


7008[8-4640659 


3875 


6667 


9042 


57 


4 


4119 


9117 


3654 


7739 


1382 


4592 


7377 


9745 


56 


5 


4873 


9863 


4392 


8470 


2106 


5309 


8086 


8-4770448 


55 


! 6 


5627 


8-4510609 


5130 


9201 


2830 


6025 


8796 


1150 


54 


' 7 


6380 


1354 


5868 


9932 


3553 


6741 


9505 


1853 


53 


8 


7133 


2100 


6607 


8-4600662 


4276 


7458 


8-4730214 


2555 


52 


9 


7887 


2846 


7344 


1393 


5000 


8174 


0923 


3257 


51 


10 


8640 


3591 


8082 


2123 


5723 


8890 


1632 


3959 


50 


11 


9393 


4336 


8820 


2853 


6446 


9605 


2341 


4661 


49 


12 


S-4470146 


5081 


9558 


3584 


7168 


8-4690321 


3050 


5363 


48 


13 


0898 


5826 


8-4560295 


4314 


7891 


1037 


3758 


6065 


47 


14 


1651 


6571 


1032 


5043 


8614 


1752 


4467 


6766 


46 


15 


2404 


7316 


1769 


5773 


9336 


2468 


5175 


7468 


45 


16 


3156 


8061 


2506 


6503 


8-4650059 


3183 


5884 


8169 


44 


17 


3908 


8805 


3243 


7232 


0781 


3898 


6592 


8871 


43 


18 


4660 


9549 


3980 


7962 


1503 


4613 


7300 


9572 


42 


19 


5412 


8-4520294 


4717 


8691 


2225 


5328 


8008 


8-4780273 


41 


20 


6164 


1038 


5453 


9420 


2947 


6043 


8715 


0974 


40 


21 


6916 


1782 


6190 


8-4610149 


3669 


6757 


9423 


1675 


39 


22 


7667 


2526 


6926 


0878 


4390 


7472 


8-4740131 


2375 


38 


23 


8419 


3269 


7662 


1607 


5112 


8186 


0838 


3076 


37 


24 


9170 


4013 


8398 


2336 


5833 


8900 


1545 


3776 


36 


25 


9921 


4757 


9134 


3064 


6555 


9615 


2253 


4477 


35 


26 


8-4480672 


5500 


9870 


3792 


7276 


8-4700329 


2960 


5177 


34 


27 


1423 


6243 


8-4570606 


4521 


7997 


1043 


3667 


5877 


33 


28 


2174 


6986 


1341 


5249 


8718 


1756 


4374 


6577 


32 


29 


2925 


7729 


2077 


5977 


9439 


2470 


5080 


7277 


31 


30 


3675 


8472 


2812 


6705 


8-4660159 


3184 


5787 


7977 


30 


31 


4426 


9215 


3547 


7433 


0880 


3897 


6494 


8677 


29 


32 


5176 


9957 


4282 


8160 


1600 


4611 


7200 


9376 


28 


33 


5926 


8 4530700 


5017 


8888 


2321 


5324 


7906 


8-4790076 


27 


34 


6676 


1442 


5752 


9615 


3041 


6037 


8612 


0775 


26 


35 


7426 


2184 


6487 


8-4620343 


3761 


6750 


9319 


1475 


25 


36 


6176 


2926 


7221 


1070 


4481 


7463 


8-4750025 


2174 


24 


37 


8925 


3668 


7956 


1797 


5201 


8176 


0730 


2873 


23 


38 


9675 


4410 


8690 


2524 


5921 


8888 


1436 


3572 


22 


39 


8'4490424 


5152 


9424 


3251 


6640 


9601 


2142 


4271 


21 


40 


1173 


5893 


3-4580158 


3978 


7360 


8-4710313 


2847 


4969 


20 


41 


1923 


6635 


0892 


4704 


8079 


1026 


3553 


5668 


19 


42 


2672 


7376 


1626 


5431 


8798 


1738 


4258 


6366 


18 


43 


3420 


8117 


2360 


6157 


9517 


2450 


4963 


7065 


17 


44 


4169 


8859 


3094 


6883 


8-4670236 


3162 


5668 


7763 


16 


45 


4918 


9599 


3827 


7609 


0955 


3874 


6373 


6461 


15 


46 


5666 


8-4540340 


4560 


8335 


1674 


4586 


7078 


9159 


14 


f 


6415 


1081 


5293 


9061 


2393 


5297 


7783 


9857 


13 


48 


7163 


1822 


6027 


9787 


3111 


6009 


8487 


8-4800555 


12 


49 


7911 


2562 


6760 


8-4630512 


3830 


6720 


9192 


1252 


11 


50 


8659 


3302 


7492 


1238 


4548 


7431 


9896 


1950 


10 


51 


9407 


4043 


8225 


1963 


5266 


8142 


8-4760600 


2648 


9 


52 


8 4500154 


4783 


8958 


2689 


5984 


8853 


1304 


3345 


8 


53 


0902 


5523 


9690 


3414 


6702 


9564 


2008 


4042 


7 


64 


1649 


6262 


8-4590422 


4139 


7420 


8-4720275 


2712 


4739 


6 


55 


2397 


7002 


1155 


4864 


8138 


0966 


3416 


5436 


5 


56 


3144 


7742 


1887 


5588 


8855 


1696 


4120 


6133 


4 


57 


3891 


8481 


2619 


6313 


9573 


2407 


4823 


6830 


3 


58 


4638 


9220 


3351 


7038 


8-4680290 


3117 


5527 


7527 


2 


59 


5385 


9960 


4082 


7762 


1008 


3827 


6230 


8223 


1 


60 


6131 


8-4550699 


. 4814 


8486 


1725 


4538 


6933 


8920 Oil 


" 


23' 


22' 


21' 


20' 


19' 


18' 


17' 


16' " 


___ 








LOG. COT AN. 88° 


• 




J 



8* 



"90 








LOG. SINE 1°. 






[Table n. | 


// 


44' 


45' 


46' ( 


47' 


48' 


49' 1 


50' I 


51' 


'' 





3-4806932 


3-4843479 


3-4S59632 


3-4930393 


3-4970784 


3-50l0798;8-5050447| 


3-5059736 


60 


1 


7628 


9163 


3-4890314 


1074 


1454 


1462 


1105 


3-5090333 


59 


2 


8323 


9357 


0997 


1750 


2124 


2126 


1762 


1040 


58 


3 


9019 


3-4S50546 


1679 


2426 


2794 


27901 


2420 


1691 


57 


4 


9714 


1235 


2361 


3102 


3463 


3453! 


3077 


2343 


56 


5 


3'4S10410 


1923 


3043 


3778 


4133 


41161 


3735 


2994 


55 


6 


1105 


2612 


3726 


4453 


4302 


4730 


4392 


3646 


54 


7 


1800 


3300 


4407 


5129 


5472 


5443I 


5049 


4297 


53 


s 


2495 


3939 


5089 


5304 


6141 


6106 


5706 


4943 


52 


9 


3190 


4677 


5771 


6430 


6310 


6769 


6363 


5599 


51 


10 


3884 


5365 


6453 


7155 


7479 


7432 


7020 


6250 


50 


11 


4579 


6053 


7134 


7830 


8143 


6095 


7677 


6901 


49 


12 


5273 


6741 


7816 


8505 


6817 


8757 


8333 


755214311 


13 


5968 


7429 


8497 


9130 


9435 


9420 


6990 


3202 


47 


14 


6662 


3116 


9173 


9355 


3-4950154 


8 -50-20032 


9646 


8353 


46 


15 


7356 


3304 


9859 


3-4940530 


0323 


0745 


3-5060303 


9503 


45 


16 


8050 


9491 


3-4900540 


1204 


1491 


1407 


0959 


3-5100154 


44 


17 


8744 


3-4S60179 


1221 


1379 


2159 


2069 


1615 


0304 


43 


18 


9433 


0866 


1902 


2553 


2327 


2731 


2271 


1454 


42 


19 


3'4820132 


1553 


2532 


3223 


3495 


3393 


2927 


2104 


41 


20 


0825 


2240 


3263 


3902 


4163 


4055 


3533 


2754 


40 


21 


1519 


2927 


3943 


4576 


4331 


4717 


4239 


3404 


39 


22 


2212 


3614 


4624 


5250 


5499 


5378 


4394 


4054 


38 


23 


2905 


4300 


5304 


5924 


6167 


6040 


5550 


4703 


37 


24 


3599 


4937 


5984 


6597 


6334 


6701 


6205 


5353136 


25 


4292 


5673 


6664 


7271 


7502 


7363 


6361 


6002135 i 


26 


4985 


6360 


7344 


7945 


8169 


6024 


7516 


6652 


34 


27 


5677 


7046 


8024 


3613 


8336 


6685 


6171 


7301 


33 


28 


6370 


7732 


8703 


9292 


9504 


9346 


8326 


7950 


32 


29 


7063 


S418 


9333 


9965 


3-4990171 


3-5030007 


9481 


8599 


31 


30 


7755 


9104 


3-4910063 


3-49o0633 


0333 


0663 


S-5070136 


9243; 30 i 


SI 


8448 


9790 


0742 


1311 


1504 


1329 


0791 


9897129 


32 


9140 


8-4870476 


1421 


1934 


2171 


1969 


1446 


8-5110546128 
1195127 


33 


9832 


1161 


2100 


2657 


2333 


2650 


2100 


34 


3-4S30524 


1847 


2779 


3330 


3504 


3310 


2755 


1343:26 


35 


1216 


2532 


3453 


4002 


4171 


3971 


3409 


2492 25 


36 


190S 


3217 


4137 


4675 


4337 


4631 


4063 


314024 


37 


2600 


3903 


4816 


5347 


5503 


5291 


4717 


378923 


38 


3291 


4533 


5495 


6020 


6169 


5951 


5371 


44371221 


39 


3983 


5273 


6173 


6692 


6835 


6611 


6025 


5035 21 1 


40 


4674 


5957 


6852 


7364 


7501 


7271 


6679 


5733 20 1 


41 


5365 


6642 


7530 


8036 


8167 


7931 


7333 


633l!l9 


42 


6057 


7327 


8208 


3703 


8333 


8590 


7937 


7029ilS 


43 


6748 


8011 


8836 


9330 


9499 


9250 


8840 


7676 17. 


44 


7439 


8696 


9564 


S-4960051 


3 -5000164 


9909 


9294 


6324116 j 


45 


8129 


9330 


S-49-20242 


0723 


0329 


5-5040569 


9947 


89721151 


46 


8820 


3-4550064 


0920 


1394 


1495 


1223 


5-5050601 


9619|l4l 


47 


9511 


0743 


1593 


2066 


2160 


1337 


1254 


5-5120266:13l 


43 


3'4,S4020r 


1432 


2275 


2737 


2325 


2546 


1907 


0914112 


49 


0892 


2116 


2953 


3403 


3490 


3205 


2560 


1561J11 


|50 


1582 


2300 


3630 


4079 


4155 


3364 


3213 


2203110 


51 


2272 


3434 


4307 


4750 


4320 


4523 


3366 


2355 


9 


52 


2962 


4167 


4934 


5421 


5435 


5131 


4513 


3502 


8 


53 


3652 


4851 


5661 


6092 


6149 


5340 


5171 


4148 


7 


54 


4342 


5534 


6333 


6763 


6314 


6493 


5323 


4795 


6 


55 


5032 


6217 


7015 


7433 


7473 


7157 


6476 


5442 


5 


56 


5721 


6900 


7692 


i 8104 


8142 


7815 


7126 


6038 


4 


57 


6411 


7533 


6363 


! 8774 


8306 


6473 


7780 


6735 


3 


58 


7100 


8266 


9045 


j 9444 


9471 


9131 


8432 


7381 


2 


5S 


7790 


8949 


9721 


34970114 


3-5010135 


9789 


9084 


8027 


1 


6C 


8479 


9632 


3-4930393 


0784 


0798 


3-5050447 


9736 


8673 







15' 


14' 


13' 


12' 


11' 


10' 


9' 


8' 


" 


L 








LOG. ( 


COSINE 8^ 


0. 




-^ J 



Table II.] 






LOG 


TAN. 1°. 




c 


)1 


// 


44' 


45' 


46' 


47' 


48' 


49' 


50' 


51' 







3-4S08920 


8-4850505 


S-4S91696 


8-4932502 


8-4972928 


8-5012982 


8-5052671 


8-5092001 


60 


1 


9616 


1195 


2380 


3179 


3598 


3646 


3329 


2653 


59 


2 


3-4810312 


1884 


3063 


3855 


4269 


4311 


3987 


3305 


58 


3 


1008 


2574 


3746 


4532 


4939 


4975 


4646 


3958 


57 


1 4 


1704 


3263 


4429 


5208 


5610 


5639 


5304 


4610 


56 


1 5 


2400 


3953 


5112 


5885 


6280 


6303 


5962 


5262 


55 


6 


3096 


4642 


5794 


6561 


6950 


6967 


6620 


5914 


54 


7 


3792 


5331 


6477 


7237 


7620 


7631 


7277 


6566 


53 


8 


4487 


6020 


7159 


7914 


8290 


8295 


7935 


7218 


52 


9 


5183 


6709 


7342 


8590 


8959 


8958 


8593 


7870 


51 


10 


5878 


7397 


8524 


9266 


9629 


9622 


9250 


8521 


50 


11 


6574 


8086 


9206 


9941 


8 4980299 


8-5020285 


9908 


9173 


49 


12 


7269 


8775 


9888 


S-4940617 


0968 


0949 


8-5060565 


9824 


48 


13 


7964 


9463 


8-4900570 


1293 


1638 


1612 


1222 


8-5100475 


47 


14 


8659 


8-4860151 


1252 


1968 


2307 


2275 


1879 


1127 


46 


15 


9353 


0839 


1934 


2643 


2976 


2938 


2536 


1778 


45 


16 


8-4820048 


1528 


2615 


3319 


3645 


3601 


3193 


2429 


44 


17 


0743 


2216 


3297 


3994 


4314 


4264 


3850 


3080 


43 


18 


1437 


2903 


3978 


4669 


4983 


4927 


4507 


3731 


42 


19 


2131 


3591 


4660 


5344 


5652 


5589 


5164 


4381 


41 


20 


2826 


4279 


5341 


6019 


6320 


6252 


5820 


5032 


40 


21 


3520 


4966 


6022 


6694 


6989 


6914 


6477 


5683 


39 


22 


4214 


5654 


6703 


7368 


7657 


7576 


7133 


6333 


38 


23 


4908 


6341 


7384 


8043 


8325 


8239 


7789 


6983 


37 


24 


5602 


7028 


8065 


8717 


8994 


8901 


8445 


7634 


36 


25 


6295 


7716 


8745 


9392 


9662 


9563 


9101 


8284 


35 


26 


6989 


8403 


9426 


8-4950066 


8-4990330 


8-5030225 


9757 


8934 


34 


27 


7682 


9089 


3-4910106 


0740 


0998 


0887 


8-5070413 


9584 


33 


28 


8376 


9776 


0787 


1414 


1666 


1548 


1069 


8-5110234 


32 


29 


9069 


8-4S70463 


1467 


2088 


2333 


2210 


1724 


0883 


31 


30 


9762 


1149 


2147 


2762 


3001 


2871 


2380 


1533 


30 


31 


8-4830455 


1836 


2827 


3435 


3668 


3533 


3035 


2183 


29 


32 


1148 


2522 


3507 


4109 


4336 


4194 


3691 


2832 


28 


33 


1841 


3209 


4187 


4783 


5003 


4855 


4346 


3482 


27 


34 


2533 


3895 


4866 


5456 


5670 


5517 


5001 


4131 


26 


35 


3226 


4581 


5546 


6129 


6337 


6178 


5656 


4780 


25 


36 


3919 


5267 


6226 


6802 


7004 


6838 


6311 


5429 


24 


37 


4611 


5952 


6905 


7476 


7671 


7499 


6966 


6078 


23 


38 


5303 


6638 


7584 


8148 


8338 


8160 


7621 


6727 


22 


39 


5995 


7324 


8263 


8821 


9005 


8821 


8275 


7376 


21 


40 


6687 


8009 


8942 


9494 


9671 


9481 


8930 


8025 


20 


41 


7379 


8695 


9621 


8-4960167 


8-5000338 


8-5040142 


9584 


8673 


19 


42 


8071 


9380 


8-4920300 


0889 


1004 


0802 


8-5080239 


9322 


18 


43 


8763 


8-4880065 


0979 


1512 


1671 


1462 


0893 


9970 


17 


44 


9454 


0750 


1658 


2184 


2337 


2122 


1547 


8-5120618 


16 


45 


8-4840146 


1435 


2336 


2856 


3003 


2782 


2201 


1267 


15 


46 


0837 


2120 


3015 


3529 


3669 


3442 


2855 


1915 


14 


47 


1528 


2805 


3693 


4201 


4335 


4102 


3509 


2563 


13 


48 


2220 


3489 


4371 


4873 


5000 


4762 


4163 


3211 


12 


49 


2911 


4174 


5049 


5544 


5666 


5421 


4817 


3859 


11 


50 


3602 


4858 


5727 


6216 


6332 


6081 


5470 


4506 


10 


51 


4292 


5543 


6405 


6888 


6997 


6740 


6124 


5154 


9 


52 


4983 


6227 


7083 


7559 


7663 


7400 


6777 


5801 


8 


53 


5674 


6911 


7761 


8231 


8328 


8059 


7430 


6449 


7 


54 


6364 


7595 


8438 


8902 


8993 


8718 


8084 


7096 


6 


i55 


7055 


8279 


9116 


9573 


9658 


9377 


8737 


7743 


5 


156 


7745 


8962 


9793 


8-4970244 


8-5010323 


8-5050036 


9390 


8391 


4 


57 


8435 


9646 


8-4930471 


0915 


0988 


0695 


8-5090042 


9038 


3 


58 


9125 


8-4890330 


1148 


1586 


1653 


1353 


0695 


9685 


2 


59 


9815 


1013 


1825 


2257 


2317 


2012 


1348 


8-5130332 III 


90 


8-4850505 


1696 


2502 


2928 


2982 


2671 


2001 


0978 Ol 




15' 


14' 


13' 


13' 


11' 


10' 


9' 


8' " 1 


j_ 


^SiSSS 






LOG. COl 


AN. 88° 






H 



92 








LOG. SINE 1°. 






{Table 11. Il 


" 


52' 


53' 


54' 


55' 


56' 


57' 


58' 


59' 


" 





8-5128673 


8-5167264 


8-5205514 


8-5243430 


8-5281017 


3-5318281 


8-5355228 


3-5391863 < 


30 


1 


9319 


7904 


6148 


4059 


1641 


8900 


5842 


2471 


59 


I 2 


9965 


8544 


6783 


4688 


2264 


9518 


6455 


3079 


58 


3 


8 5130611 


9184 


7417 


5317 


2888 


3-5320136 


7068 


3687 


57 


4 


1256 


9824 


8052 


5946 


3511 


0754 


7680 


4295 


56 


5 


1902 


8-5170464 


8686 


6574 


4135 


1372 


8293 


4902 


55 


6 


2548 


1104 


9320 


7203 


4758 


1990 


8906 


5510 


54 


7 


3193 


1743 


9954 


7833 


5381 


2608 


9518 


6117 


53 


8 


3838 


2383 


8-5210588 


8460 


6004 


3226 


8-5360131 


6725 


52 


9 


4484 


3023 


1222 


9088 


6627 


3844 


0743 


7332 


51 


10 


5129 


3662 


1856 


9717 


7250 


4461 


1356 


7939 


50 


11 


5774 


4301 


2490 


8-5250345 


7873 


5079 


1968 


8546 


49 


12 


6419 


4941 


3123 


0973 


8495 


5696 


2580 


9153 


48 


13 


7064 


5580 


3757 


1601 


9118 


6313 


3192 


9760 


47 


14 


7708 


6219 


4390 


2229 


9741 


6931 


3804 


8-5400367 


46 


15 


8353 


6858 


5024 


2857 


8-5290363 


7548 


4416 


0974 


45 


16 


8997 


7497 


5657 


3485 


0985 


8165 


5028 


1581 


44 


17 


9642 


8135 


6290 


4112 


1608 


8782 


5640 


2187 


43 


18 


3-5140286 


8774 


6923 


4740 


2230 


9399 


6251 


2794 


42 


19 


0931 


9413 


7556 


5367 


2852 


8-5330015 


6863 


3400 


41 


20 


1575 


8-5180051 


8189 


5995 


3474 


0632 


7474 


4007 


40 


21 


2219 


0689 


8822 


6622 


4096 


1249 


8086 


4613 


39 


22 


2863 


1328 


9455 


7249 


4718 


1865 


8697 


5219 


38 


23 


3507 


1966 


8-5220087 


7877 


5339 


2482 


9308 


5825 


37 


24 


4150 


2604 


0720 


8504 


5961 


3098 


9920 


6431 


36 


25 


4794 


3242 


1352 


9131 


6583 


3714 


8-5370531 


7037 


35 


26 


5438 


3880 


1985 


9757 


7204 


4330 


1142 


7643 


34 


27 


6081 


4518 


2617 


8-5260384 


7826 


4946 


1752 


8249 


33 


28 


6725 


5156 


3249 


1011 


8447 


5562 


2363 


8854 


32 


29 


7368 


5793 


3681 


1637 


9068 


6178 


2974 


9460 


31 


30 


8011 


6431 


4513 


2264 


9689 


6794 


3585 


8-5410066 


30 


31 


8654 


7068 


5145 


2890 


8-5300310 


7410 


4195 


0671 


29 


32 


9297 


7706 


5777 


3517 


0931 


8026 


4806 


1276 


28 


33 


9940 


8343 


6408 


4143 


1552 


8641 


5416 


1882 


27 


34 


3-5150583 


8980 


7040 


4769 


2173 


9257 


6026 


2487 


26 


35 


1226 


9617 


7672 


5395 


2793 


9872 


6636 


3092 


25 


36 


1869 


8-5190254 


8303 


6021 


3414 


8-5340487 


7247 


3697 


24 


37 


2511 


0891 


8934 


6647 


4034 


1103 


7857 


4302 


23 


38 


3154 


1528 


9566 


7273 


4655 


1718 


8466 


4907 


22 


39 


3796 


2164 


8-5230197 


7898 


5275 


2333 


9076 


5511 


21 


40 


4438 


2801 


0828 


8524 


5895 


2948 


9686 


6116 


20 


41 


5080 


3438 


1459 


9149 


6516 


3563 


8-5380296 


6721 


19 


42 


5722 


4074 


2090 


9775 


7136 


4177 


0905 


7325 


18 


43 


6364 


4710 


2720 


8-5270400 


7756 


4792 


1515 


7929 


17 


44 


7006 


5347 


3351 


1025 


8375 


5407 


2124 


8534 


16 


45 


7648 


5983 


3982 


1651 


8995 


6021 


2734 


9138 


15 


46 


8290 


6619 


4612 


2276 


9615 


6636 


3343 


9742 


14 


47 


8931 


7255 


5243 


2901 


8-5310235 


.7250 


3952 


8-5420346 


13 


48 


9573 


7891 


5873 


3525 


0854 


7864 


4561 


0950 


12 


49 


8-5160214 


8526 


6503 


4150 


1473 


8478 


5170 


1554 


11 


50 


0856 


9162 


7133 


4775 


2093 


9092 


5779 


2158 


10 


51 


1497 


9798 


7763 


5400 


2712 


9706 


6388 


2762 


9 


52 


2138 


8-5200433 


8393 


6024 


3331 


8-5350320 


6997 


3365 


8 


53 


2779 


1069 


9023 


6648 


3950 


0934 


7605 


3969 


7 


54 


3420 


1704 


9653 


7273 


4569 


1548 


8214 


4572 


6 


55 


4061 


2339 


8-5240283 


7897 


5188 


2161 


8822 


5176 


5 


56 


4701 


2974 


0912 


8521 


5807 


2775 


9431 


5779 


4 


57 


5342 


3609 


1542 


9145 


6426 


3389 


8-5390039 


6382 


3 


58 


5983 


4244 


2171 


9769 


7044 


4002 


0647 


6986 


2 


59 


6623 


4879 


2800 


8-52^0393 


7663 


4615 


1255 


7589 


1 


60 


7264 


5514 


3430 


1017 


8281 


5228 


1863 


- 8192 





ft 


7' 


6' 


5' 


4' 


3' 


2' 


1' 


C 


" 


__ 








LOG. CO 


SINE 88° 






'\ 



Table II.] 






LOG. TAN. 1°. 






93 ll 


" 


52' 


53' 


54' 


55' 


56' 


57' 


58' 


59' 


// 


C 


3-513097S 


3-5169610 


8-5207902 


8-5245860 


8-5283490 


3-5320797 


8-5357787 


8-5394466 


60 


1 


1625 


3-5170251 


8537 


6490 


4114 


1416 


8401 


5075 


59 


2 


2272 


0892 


9173 


7120 


4739 


2035 


9015 


5683 


58 


3 


2918 


1533 


9808 


7749 


5363 


2654 


9629 


6292 


57 


4 


3564 


2173 


8-5210443 


8379 


5987 


3273 


8-5360242 


6900 


56 


5 


4211 


2814 


1078 


9008 


6611 


3892 


0856 


7509 


55 


6 


4857 


3455 


1713 


9638 


7235 


4510 


1469 


8117 


54 


7 


5503 


4095 


2348 


8-5250267 


7859 


5129 


2082 


8725 


53 


8 


6149 


4735 


2982 


0896 


8483 


5747 


2696 


9333 


52 


9 


6795 


5375 


3617 


1525 


9106 


6366 


3309 


9941 


51 


10 


7441 


6016 


4251 


2154 


9730 


6984 


3922 


8-5400549 


50 


11 


8087 


6656 


4886 


2783 


8-5290353 


7602 


4535 


1157 


49 


12 


8732 


7296 


5520 


3412 


0977 


8220 


5148 


1765 


48 


13 


9378 


7935 


6154 


4041 


1600 


8838 


5761 


2372 


47 


14 


8-5140023 


8575 


6789 


4669 


2223 


9456 


6373 


2980 


46 


15 


0668 


9215 


7423 


5298 


2847 


8-5330074 


6986 


3587 


45 


16 


1314 


9854 


8057 


5926 


3470 


0692 


7599 


4195 


44' 


17 


1959 


8-5180494 


8690 


6555 


4093 


1310 


6211 


4802 


43 


18 


2604 


1133 


9324 


7183 


4716 


1927 


8823 


5409 


42 


19 


3249 


1772 


9958 


7811 


5338 


2545 


9436 


6017 


41 


20 


3894 


2412 


8-5220591 


8439 


5961 


3162 


8-5370048 


6624 


40 


21 


4539 


3051 


1225 


9067 


6584 


3779 


0660 


7231 


39 


22 


5183 


3690 


1858 


9695 


7206 


4397 


1272 


7838 


38 


23 


5828 


4329 


2492 


8-5260323 


7829 


5014 


1884 


8445 


37 


24 


6472 


4967 


3125 


0951 


8451 


5631 


2496 


9051 


36 


25 


7117 


5606 


3758 


1579 


9073 


6248 


3108 


9658 


35 


26 


7761 


6245 


4391 


2206 


9696 


6865 


3719 


8-5410264 


34 


27 


8405 


6883 


5024 


2834 


8-5300318 


7482 


4331 


0871 


33 


28 


9049 


7522 


5657 


3461 


0940 


8098 


4942 


1477 


32 


29 


9693 


8160 


6290 


4088 


1562 


8715 


5554 


2084 


31 


30 


S-5150337 


8798 


6922 


4716 


2183 


9331 


6165 


2690 


30 


31 


0981 


9436 


7555 


5343 


2805 


9948 


6777 


3296 


29 


32 


1625 


8-5190074 


8187 


5970 


3427 


8-5340564 


7388 


3902 


28 


33 


2268 


0712 


8820 


6597 


4048 


1181 


7999 


4508 


27 


34 


2912 


1350 


9452 


7223 


4670 


1797 


8610 


5114 


26 


35 


3555 


1988 


8-5230084 


7850 


5291 


2413 


9221 


5720 


25 


36 


4199 


2626 


0717 


8477 


5912 


3029 


9832 


6326 


24 


37 


4842 


3263 


1349 


9103 


6534 


3645 


8-5380442 


6931 


23 


38 


5485 


3901 


1980 


9730 


7155 


4261 


1053 


7537 


22' 


39 


6128 


4538 


2612 


8-5270356 


7776 


4876 


1664 


8142 


21; 


40 


6771 


5175 


3244 


0983 


8397 


5492 


2274 


8748 


201 


41 


7414 


5813 


3876 


1609 


9018 


6108 


2884 


9353 


191 


42 


8057 


6450 


4507 


2235 


9638 


6723 


3495 


9958 


181 


43 


8699 


7087 


5139 


2861 


8-5310259 


7339 


4105 


8-5420563 


171 


44 


9342 


7724 


5770 


3487 


0880 


7954 


4715 


1168 


16 


45 


9984 


8361 


6401 


4113 


1500 


8569 


5325 


1773 


15 


46 


8-5160627 


8997 


7033 


4739 


2121 


9184 


5935 


2378 


14 


47 


1269 


9634 


7664 


5364 


2741 


9799 


6545 


2983 


13 


48 


1911 


8-5200271 


8295 


5990 


3361 


8-5350414 


7155 


3588 


12 


49 


2553 


0907 


8926 


6615 


3981 


1029 


7765 


4193 


11 


50 


3195 


1543 


9557 


7241 


4601 


1644 


8374 


4797 


10 


51 


3837 


2180 


8-5240187 


7866 


5221 


2259 


8984 


5402 


9 


52 


4479 


2816 


0818 


8491 


5841 


2873 


9593 


6006 


8 


53 


5121 


3452 


1449 


9116 


6461 


3488 


8-5390203 


6610 


7 


54 


5762 


4088 


2079 


9741 


7081 


4102 


0812 


7214 


6 


55 


6404 


4724 


2709 


8-5280366 


7700 


4717 


1421 


7819 


5 


56 


7045 


5360 


3340 


0991 


8320 


5331 


2030 


8423 


4 


57 


7687 


5995 


3970 


1616 


8939 


5945 


2639 


9027 


3 


58 


8328 


6631 


4600 


2241 


9559 


6559 


3248 


9631 


2 


59 


8969 


7267 


523,0 


2865 


3-5320178 


7173 


3857 


3-5430234 


1 


60 


9610 


7902 


5860 


3490 


0797 


7787 


4466 


0838 





// 


7' 


6' 


5' 


4' 


3' 


2' 


1' 


0' 1 


" 


SiS 






L 


3G. COT-A 


N. 88o. 






^1 



94 



LOG. SINE. 



[Table II. 



4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
|42 
|43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60, 



2° 

S'5428192 
64218 
99948 

S-5535386 
70536 
•5605404 
39994 
74310 
•5708357 
42139 
75660 

8-5808923 
41933 
74694 

S-5907209 
39483 
71517 

S-6003317 
34886 
66226 
97341 
5128235 
58910 
89369 

8-6219616 
49653 
79484 

363091 11 
38537 
67764 

96796 

8-6425634 

54282 

82742 

3-6511016 

39107 

67017 

94748 

6622303 

49684 

76893 

8-6703932 

30804 

57510 

84052 

8-6810433 

36654 

62718 

88625 

8-6914379 

39980 

65431 

90734 

7015889 

40899 

65766 

90490 

8-7U5075 

39520 

63829 

88002 

87^ 



diff. 

36026 
35730 
35438 
35150 
34868 
34590 
34316 
34047 
33782 
33521 

33263 
33010 
32761 
32515 
32274 
32034 
31800 
31569 
31340 
31115 
30894 
30675 
30459 
30247 
30037 
29831 
29627 
29426 
29227 
29032 

23838 
23648 
28460 
28274 
28091 
27910 
27731 
27555 
27381 
27209 
27039 
26672 
26706 
26542 
26381 
26221 
26064 
25907 
25754 
25601 
25451 
25303 
25155 
25010 
24867 
24724 
24585 
24445 
24309 
24173 



3° 

8-7188002 

8-7212040 
35946 
59721 
83366 
7306882 
30272 
53535 
76675 
99691 

S-7422586 
45360 
68015 
90553 

8-7512973 
35278 
57469 
79546 

3-7601512 
23366 
45111 
66747 
88275 

3-7709697 
31014 
52226 
73334 
94340 

8-7815244 
36043 
56753 
77359 
97867 

3-7918278 
33594 
58314 
78941 
98974 

3'8018915 
38764 

58523 
78192 
97772 

3-8117264 
36668 
55985 
75217 
94363 

3-8213425 
32404 

51299 
70112 
8S844 

8-8307495 
26066 
44557 
62969 
81304 
99561 

8-8117741 
35845 
86^ 



diff. 

•24038 
23906 
23775 
23645 
23516 
23390 
•23263 
23140 
•23016 
22395 
22774 
22655 
22538 
22420 
22305 
22191 
22077 
21966 
21854 
21745 
21636 
21528 
21422 
21317 
21212 
21108 
21006 
20904 
20804 
20705 
20606 
20508 
20411 
20316 
20220 
20127 
20033 
19941 
19849 
19759 
19669 
19530 
19492 
19404 
19317 
19232 
19146 
19062 
18979 
18895 

18813 
18732 
18651 
13571 
18491 
18412 
1833, 
18257 
18180 
18104 



40 

8'S435845 
53874 
71827 
89707 

8^8507512 
25245 
42905 
60493 
78010 
95457 

[•S612333 
30139 
47376 
64545 
81646 



3-8715646 
32546 
49381 
66150 
82854 
99493 

3-8816069 
32531 
49031 
65418 
81743 
98007 

S-S914209 
30351 
46433 
62455 
73418 
94322 

3-9010168 
25955 
41685 
57353 
72975 
8S535 

8-9104039 
19437 
34831 
50219 
65504 
80734 
95911 

3-9211034 
26105 
41123 
56089 
71003 
85866 

3-9300678 
15439 
30150 
44811 
59422 
73983 
83496 

3-9402960 
85=> 



diff. 

18029 

17953 

17 

17805 

17733 

17660 

17588 

17517 

17447 

17376 

17306 
17237 
17169 
17101 
17034 
16966 
16900 
16835 
16769 
16704 
16639 
16576 
16512 
16450 
16387 
16325 
16264 
16202 
16142 
16082 
16022 
15963 
15904 
15346 
15787 
15730 
15673 
15617 
15560 
15504 
15448 
15394 
15333 
15285 
15230 
15177 
15123 
15071 
15018 
14966 
14914 
14363 
14812 
14761 
14711 
14661 
14611 
14561 
14513 
14464 



diff 

LOG. COSINE. 



5<^ 
9402960 
17376 
31743 
46063 
60335 
74561 
88739 
3-9502871 
16957 
30996 

44991 
58940 
72843 
86703 
9600517 
14288 
23014 
41697 
55337 
68934 
82487 
95999 

3-9709463 
22395 
36280 
49624 
62926 
76188 
89408 

3-9302589 
15729 
28329 
41889 
54910 
67391 
80834 
93737 

3-9906602 
19429 
32217 
44968 
57631 
70356 
82994 
95595 
•0008160 
20687 
33179 
45634 
58053 

70436 
82784 
95096 
9^0107374 
19616 
31323 
43996 
56135 
68239 
80309 
92346 
84^ 



diff 

14416 
14367 
14320 
14272 
14226 
14178 
14132 
14086 
14039 
13995 

13949 
13903 
13860 
13814 
13771 
13726 
13633 
13640 
13597 
13553 
13512 
13469 
13427 
13385 
13344 
13302 
13262 
13220 
13181 
13140 
13100 
13060 
13021 
12981 
12943 
12903 
12865 
12827 
12788 
12751 
12713 
12675 
12633 
12601 
12565 
12527 
12492 
12455 
12419 
12333 

12348 

1-2312 

1227 

12242 

12207 

12173 

12139 

12104 

12070 

12037 

diff. 



6° 

•0192346 

■0204348 
16318 
28254 
40157 
52027 
63865 
75669 
87442 
99182 

•0310890 
22567 
34212 
45825 
him 
63958 
80477 
91966 
90403424 
14852 
26249 
37617 
48954 
60261 
71538 
82786 
94005 
9-0505194 
16354 
27435 
33588 
49661 
60706 
71723 
82711 
93672 

■0604604 
15509 
26336 
37235 
48057 
58852 
69619 
80360 
91074 
9 0701761 
12421 
23055 
33663 
44244 

54799 
65329 
75832 
86310 
86762 
9-0S07189 
17590 
27966 
33317 
48643 
53945 
83° 



diff 

12002 
11970 
11936 
11903 
11370 
11338 
11804 
11773 
11740 
11708 
11677 
11645 
11613 
11582 
11551 
11519 
11489 
11453 
114-28 
11397 
11368 
11337 
11307 
11277 
11248 
11219 
11189 
11160 
11131 
11103 
11073 
11045 
11017 
10983 
10961 
10932 
10905 
10877 
10849 
10322 

10795 
10767 
10741 
10714 
10687 
10660 
10634 
10608 
10581 
10555 

10530 
10503 
10478 
10452 
10427 
10401 
10376 
10351 
10326 
10302 

diff. 



Table II.] 



LOG. TAN. 



95 



03-&130338 
1 66909 
2S-5502683 



38166 
73362 

3-5608276 
42912 
77275 

8-5711368 
45197 

78766 
3-5812077 
45136 
77945 
S-5910509 
42332 
74917 
S-6006767 
38386 
69777 

8.6100943 
31889 
62616 
93127 

8-6223427 
53518 
83402 

8-6313033 
42563 
71845 
6400931 
29825 
58528 
87044 

8-6515375 
43522 
71490 
99279 

8-6626891 
54331 
81598 

3-6703697 
35628 
62393 



S-6815437 

41719 

67844 

93813 

6919629 

45292 
70806 
96172 

8-7021390 
46465 
71395 
96135 

8-7120834 
45345 
69719 
93958 



Mjf. 

36071 
35774 
35483 
35196 
34914 
34636 
34363 
34093 
33829 
33569 



3« 

8-7I9395S 

3-7213063 
42035 
65877 
89589 

13.7313174 
36631 
59964 
83172 

3-7406253 



33311 

33059 
32809 
32564 
32323 
32085 
31850 
31619 
31391 
31166 

30946 
30727 
30511 
30300 
30091 
29834 
29631 
29430 
29232 
29086 



23394 
23703 
28516 
28331 
28147 
27968 
27789 
27612 
27440 
27267 

27099 
26931 
26765 
26603 
26441 
26282 
26125 
25969 
25816 
25663 
25514 
25366 
25213 
25075 
24930 
24790 
24649 
24511 
2437 
24239 

diff. 



29222 
52067 
74792 
97400 

S-7519892 
42269 
64531 
86631 

3-7603719 
30647 

52465 
74175 
95777 

3-7717274 
33665 
59952 
81136 

3-7802213 
23199 
44079 

64861 
35544 

3-7906130 
26620 
47014 
67313 
87519 
8007632 
27653 
47533 
67422 
87172 

3-8106334 
26407 
45994 
65294 
84608 

S-8203838 
22984 
42046 

G1026 
79924 
98741 

3-8317478 
36134 
54712 
7.3211 
91633 

3-3409977 
28245 
4C437 
86^ 



diff. I 4 
.,4,nrJ3-S445437 
:4J^2 64554 
•^■^'^ 82597 
3-8500566 
18461 
36283 
54034 
71713 
89321 
3-8606859 



23342 
23712 
23585 
23457 
23333 
23208 
23036 
22964 

22845 
22725 
22603 
22492 
22377 
22262 
22150 
22038 
21923 
21318 

21710 
21602 
2149' 
21391 

21287 
21184 
21082 
20931 
20330 
20732 

20633 
20586 
20490 
20394 
20299 
20206 
0113 
20021 
19930 
19839 
19750 
19662 
19573 
19487 
19400 
19314 
19230 
19146 
19062 



18817 
18737 
18656 
18578 
18499 
18422 
18344 
18268 
18192 



diff 

18117 

18043 
17969 
17395 
17322 
17751 
17679 
17608 
17538 
17463 

17398 
17330 
17262 
17194 
17127 
17061 
18995 
16929 
16864 
16799 
16736 
16672 
16609 
16547 
16484 
16423 
16362 
18301 
16240 
16182 

16121 
16083 
16004 
15947 
15889 
15831 
15775 
15718 
15663 
15607 
15552 
15497 
15443 
15388 
15335 
15282 
15229 
15177 
15124 
15073 
15021 
14971 
14919 
14869 
14320 
14769 
14721 
14671 
14623 
14574 

dij. 

LOG. COTAN. 



24327 
41725 
59055 
76317 
93511 
8-8710633 
27699 
44694 
61623 
78437 

95285 

38812022 
28694 
45303 
61850 
78334 
94757 

3-3911119 
27420 
43660 
59842 
75963 
9202^ 

3-9003030 
23977 
39866 
55697 
71472 
87190 

3-9102853 
18460 
34012 
49509 
64952 
80340 
95675 

3-9210957 
26186 
41363 
56487 

71560 
86531 

8-9301552 
16471 
31340 
46160 
60929 
75650 
90321 

3-9404944 
19518 
85° 



5° 

•9419518 
34044 
43523 
62954 
77338 
91676 

•9505967 
20211 
34410 
48564 
62672 
76735 
90754 

-9604728 
18659 
32545 
46388 
60183 
73944 
87653 

-9701330 
14959 
28547 
42092 
55597 
69060 
82483 
95865 

-9809208 
22507 

35769 
48991 
62173 
75317 
88421 
9901487 
14514 
27503 
40454 
53367 

66243 
79081 
91883 
•0004647 
17375 
30066 
42721 
55340 
67924 
80471 

92984 
-0105461 
17903 
30310 
42682 
55021 
67325 
79594 
91831 
■0204033 
16202 
84^ 



diff 

14526 
14479 
14431 
14384 
14338 
14291 
14244 
14199 
14154 
14108 

14063 
14019 
13974 
13931 
13888 
13343 
13800 
13756 
13714 
13672 

13629 
13588 
13545 
13505 
13463 
13423 
13382 
13341 
13301 
13262 

13222 
13182 
13144 
13104 
13066 
13027 
12989 
12951 
12913 
12876 

12838 
12802 
12764 
12723 
12691 
12655 
12619 
12584 
12547 
12513 
12477 
12442 
12407 
12372 
12339 
12304 
12269 
12237 
12202 
12169 

diff. 



6^ 

9-0216202 
28338 
40441 
52510 
64548 
76552 
88524 
0300464 
12373 
24249 

36093 
47906 
59683 
71439 
83159 
94848 
9-0406506 
18134 
29731 
41299 
52836 
64343 
75821 
87270 



9-0510078 
21439 
32771 
44074 
55349 
66595 
77813 
89002 

9-0600164 
11297 
22403 
33482 
44533 
55556 
66553 

77522 
88465 
99381 
9-0710270 
21133 
31969 
42779 
53563 
64321 
75053 

85760 
96441 
•0807096 
17726 
28331 
38911 
49466 
59996 
70501 
80981 
91438 
83° 



diff 

12136 
12103 
12069 
12038 
12004 
11972 
11940 
11909 
11876 
11844 

11813 
11782 
11751 
11720 
11689 
11658 
11828 
11597 
11568 
11537 
11507 
11478 
11449 
11419 
11389 
11361 
11332 
11303 
11275 
11246 
11218 
11189 
lil62 
11133 
11106 
11079 
11051 
11023 
10997 
10969 

10943 
10916 
10889 
10863 
10836 
10810 
10784 
10758 
10732 
10707 

10681 
10655 
10630 
10605 
10580 
10555 
10530 
10505 
10480 
10457 

diff 



96 



LOG. SINE. 



[Table n. 



9-0358945 
69221 
79473 
89700 
99903 

9-0910082 
20237 
30367 
40474 
50556 
60615 
70651 
80662 
80651 

9-1000616 
10558 
20477 
30373 
40246 
50096 
59924 
69729 
79512 
89272 
99010 
•1108726 
18420 
28092 
37742 
47370 

56977 
66562 
76125 
85667 
95188 

9-1204688 
14167 
23624 
33061 
42477 
51872 
61246 
70600 
79934 
89247 
98539 

9-1307812 
17064 
26297 
35509 
44702 
53875 
63028 
72161 
81275 
90370 
99445 

9-1408501 
17537 
26555 
35553 



diff. 

10276 
10252 
10227 
10203 
10179 
10155 
10130 
10107 
10082 
10059 
10036 
10011 
9989 
9965 
9942 
9919 
9896 
9873 
9850 
9828 

9805 
9783 
9760 
9738 
9716 
9694 
9672 
9550 
9628 
9607 

9585 
9563 
9542 
9521 
9500 
9479 
9457 
9437 
9416 
9395 
9374 
9354 
9334 
9313 
9292 
9273 
9252 
9233 
9212 
9193 

9173 
9153 
9133 
9114 
9095 
9075 
9056 
9036 
9018 



8° 

•1435553 
44532 
53493 
62435 
71358 
80262 
89148 
98015 

•1506864 
15694 
24507 
33301 
42076 
50834 
59574 
68296 
77000 
85686 
94354 

•1603005 

11639 
20254 
28853 
37434 
45998 
54544 
63074 
71586 
80081 
88559 
97021 

•1705465 
13893 
22305 
30699 
39077 
47439 
55784 
64112 
72425 
80721 
89001 
97265 

•1805512 
13744 
21960 
30160 
38344 
46512 
54665 
62802 
70923 
79029 
87120 
95195 

1-1903254 
11299 
19328 
27342 
35341 
43324 
81<^ 



diff. 

8979 
8961 
8942 
8923 
8904 
8886 
8867 
8849 
8830 
8813 
8794 
8775 
8758 
8740 
8722 
8704 



8651 
8634 

8615 
8599 
8581 
8564 
8546 
8530 
8512 
8495 
8478 
8462 
8444 
8428 
8412 
8394 
8378 
8362 
8345 
8328 
8313 
8296 
8280 
8264 
8247 
8232 
8216 
8200 
8184 
8168 
8153 
8137 
8121 
8106 
8091 
8075 
8059 
8045 
8029 
8014 
7999 
7983 



9° 

19-1943324 
51293 
59247 
67186 
75110 
83019 
90913 
98793 

9-2006658 
14509 

22345 
30167 
37974 
45766 
53545 
61309 
69059 
76795 
84516 
92224 

99917 
9-2107597 
15263 
22914 
30552 
38176 
45787 
53384 
60967 
68536 
76092 
83635 
91164 



7969 
7954 
7939 
7924 
7909 
7894 
7880 
7865 
7851 
7836 

7822 
7807 
7792 
7779 
7764 
7750 
7736 
7721 
7708 
7693 

7680 
7666 
7651 
7638 
7624 
7611 
7597 
7583 
7569 
7556 
7543 
7529 
7516 
7502 
7489 
7476 
7462 
7450 
7436 
7423 

7410 
7397 
7385 
7371 
7358 
7346 
7333 
7320 
7307 
7295 
7282 
7270 
7257 
7245 
7232 
7220 
7207 
7196 
7183 
7170 

diff. 
LOG. COSINE. 



9-2206182 
13671 
21147 
28609 
36059 
43495 
50918 
58328 
65725 
73110 
80481 
87839 
95185 

9-2300518 
09838 
17145 

24440 
31722 
38992 
46249 
53494 
60726 
67946 
75153 
82349 
89532 
96702 
SQo 



IQJ 

9-2396702 
9-240386] 
11007 
18141 
25264 
32374 
39472 
46558 
53632 
60695 

67746 
74784 
81811 
88827 
95830 
9-2502822 
09803 
16772 
23729 
30675 

37609 
44532 
51444 
58344 
65233 
72110 
78977 
85832 
92676 
99509 
9-2606330 
13141 
19941 
26729 
33507 
40274 
47030 
53775 
60509 
67232 

73945 
80647 
87338 
94019 
9-2700689 
07348 
13997 
20635 
27263 
33880 
40487 
47083 
53669 
60245 
66811 
73366 
79911 
86445 
92970 
99484 



79<= 



7159 
7146 
7134 
7123 
7110 
7098 
7086 
7074 
7063 
7051 
7038 
7027 
7016 
7003 
6992 
6981 
6969 
6957 
6946 
6934 

6923 
6912 
6900 
6889 
6877 
6867 
6855 
6844 
6833 
6821 

6811 
6800 
6788 
6778 
6767 
6756 
6745 
6734 
6723 
6713 
6702 
6691 
6681 
6670 
6659 
6649 
6638 
6628 
6617 
6607 

6596 
6586 
6576 
6566 
6555 
6545 
6534 
6525 
6514 
6504 

diff. 



11° 

-2805988 
12483 
18967 
25441 
31905 
38359 
44803 
51237 
57661 
64076 
70480 
76875 
83260 
89636 
96001 

-2902357 
08704 
15040 
21367 
27685 
33993 
40291 
46580 
52859 
59129 
65390 
71641 
77883 
84116 
90339 
96553 

-3002758 
08953 
15140 
21317 
27485 
33644 
39794 
45934 
52066 

58189 
64303 
70407 
76503 
82590 
88668 
94737 
)-3100798 
06849 
12892 

18926 
24951 
30968 
36976 
42975 
48965 
54947 
60921 
66885 
72841 
78789 
78° 



dif. 

6495 
6484 
6474 
6464 
6454 
6444 
6434 
6424 
6415 
6404 

6395 
6385 
6376 
6365 
6356 
6347 
6336 
6327 
6318 
6308 
6298 
6289 
6279 
6270 
6261 
6251 
6242 
6233 
6223 
6214 

6205 
6195 
6187 
6177 
6168 
6159 
6150 
6140 
6132 
6123 
6114 
6104 
6096 
6087 
6078 
6069 
6061 
6051 
6043 
6034 

6025 
6017 
6008 
5999 
5990 
5982 
5974 
5964 
5956 
5948 



Table ii.] 



LOG. TAN. 



97 



7° I diff. I 8° 
90891438L4Q1 9-1478025 
90901869 ln.no 87182 
96321 



12277 
22660 
33020 
43355 
53667 
63955 
74219 
84460 
94678 
9.1004872 
15044 
25192 
35317 
45420 
55500 
65557 
75591 
85604 

95594 
91105562 
15508 
25431 
35333 
45213 
' 55072 
64909 
74724 
84518 
94291 
9-1204043 
13773 
23482 
33171 
42839 
52486 
62112 
71718 
81303 



9.1300413 
09937 
19442 
28926 
38391 
47835 
57260 
66665 
76051 
85417 
94764 

9-1404092 
13400 
22689 
31959 
41210 
50442 
59655 
68849 
78025 
820 



10408 
10383 
10360 
10335 
10312 
10288 
10264 
10241 
10218 
10194 
10172 
10148 
10125 
10103 
10080 
10057 
10034 
10013 
9990 

9968 
9946 
9923 
9902 
9880 
7859 
9837 
9815 
9794 
9773 
9752 
9730 
9709 
9689 
9668 
9647 
9626 
9606 
9585 
9565 

9545 

9524 

9505 

9484 

9465 

9444 

9425 

9405 

9386 

9366 

934 

9328 



9-1505441 
14543 
23627 
32692 
41739 
50769 
59780 
68773 
77748 
86706 
95646 

9-1604569 
13473 
22361 
31231 
40083 
48919 
57737 
66538 
75322 



9157 
9139 
9120 
9102 
9084 
9065 
9047 
9030 
9011 
8993 

8975 
9958 
8940 
8923 
8904 



9289 
9270 
9251 
9232 
9213 
9194 
9176 

diff. 



92839 
9-1701572 
10289 
18989 
27672 
36338 

44988 
53622 
62239 
70840 
79425 
87993 
96546 

9-1805082 
13602 
22106 
30595 
39068 
47525 
55966 
64392 
72802 
81196 
89575 
97939 

9-1906287 

14621 
22934 
31241 
39529 
47802 
56059 
64302 
72530 
80743 
88941 
97125 
Slo 



8870 
8852 
8836 
8818 
8801 
784 
8767 
8750 
8733 
8717 
8700 
8683 
8666 



9° 
9-1997125 
9-2005294 
13449 
21588 
29714 
37825 
45922 
54004 
62072 
70126 
78165 
86191 
94203 
9-2102200 
10184 
18153 
26109 
34051 
41980 
49894 

57795 
65683 
73556 
81417 
89264 
97097 
2204917 
12724 
20518 
28298 



8634 
8617 
8601 
8585 
8568 
8553 
8536 
8520 
8504 
8489 
8473 
8457 
8441 
8426 
8410 
8394 
8379 
8364 
8348 
8334 

8318 
8302 
8288 
8273 
8257 
8243 
8228 
8213 
8198 
8184 



36065 
43819 
51561 
59289 
67004 
74706 
82395 
90071 
97735 

9-2305386 
13024 
20650 
28262 
35863 
43451 
51026 
58589 
66139 
73678 
81203 
88717 
96218 

9-2403708 
11185 
18650 
26103 
33543 
40972 
48389 
55794 
63188 
80° 



8169 
8155 
8139 
8126 
8111 
8097 
8082 
8068 
8054 
8039 
8026 
8012 
7997 
7984 
7969 
7956 
7942 
7929 
7914 
7901 
7888 
7873 
7861 
7847 
7833 
7820 
7807 
7794 
7780 
7767 

7754 
7742 
77-28 
7715 
7102 
7689 
7676 
7664 
7651 
7638 

7626 
7612 
7601 
7588 
7575 
7563 
7550 
7539 
7525 
7514 

7501 
7490 
7477 
7465 
7453 
7440 
7429 
7417 
7405 
7394 



10° 

2463188 
70569 
77939 
85297 
92643 
99978 

9-2507301 
14612 
21912 
29200 
36477 
43743 
50997 
58240 
65472 
72692 
79901 
87099 
94285 

9-2601461 
08625 
15779 
22921 
30053 
37173 
44283 
51382 
58470 
65547 
72613 

79669 
86714 
93749 
9-2700772 
07786 
14788 
21780 
28762 
35733 
42694 



diff. 



49644 
56584 
63514 
70434 
77343 
84242 
91131 
98009 
9-2804878 
11736 

18585 
25423 
32251 
39070 
45878 
52677 
59466 
66245 
73014 
79773 
86523 
79° 



7381 
7370 
7358 
7346 
7335 
7323 
7311 
7300 
7288 
7277 

7266 
7254 
7243 
7232 
7220 
7209 
7198 
7186 
7176 
7164 
7154 
7142 
7132 
7120 
7110 
7099 
7088 
7077 
7066 
7056 
7045 
7035 
7023 
7014 
7002 
6992 
6982 
6971 
6961 
6950 
6940 
6930 
6920 
6909 
6899 
6889 
6878 
6869 



6649 



6828 
6819 
6808 
6799 
6789 
6779 
6769 
6759 
6750 

diff. 



11° i 
9-2886523 
93263 
99993 
9-2906713 
13424 
20126 
26817 
33500 
40172 
46836 

53489 
60134 
66769 
73395 
80011 
86618 
93216 
99804 
9-3006383 
12954 
19514 
26066 
32609 
39143 
45667 
52183 
58689 
65187 
71675 
78155 

84626 
91088 
97541 

9-3103985 
10421 
16848 
23266 
29675 
36076 
42468 
48851 
55226 
61592 
67950 
74299 
80640 
86972 
93295 
99611 

9-3205918 

12216 
18506 
24788 
31061 
37327 
43584 
49632 
56073 
62305 
68529 
74745 
78° 



6740 
6730 
6720 
6711 
6702 
6691 
6683 
6672 
6664 
6653 
6645 
6635 
6626 
6616 
6607 
6598 
6588 
6579 
6571 
6560 
6552 
6543 
6534 
6524 
6516 
6506 
6498 
6488 
6480 
6471 
6462 
6453 
6444 
6436 
6427 
6418 
6409 
6401 
6392 
6383 

6375 
6366 
6358 
6349 
6341 
6332 
6323 
6316 
6307 
6296 

6290 
6282 
6273 
6266 
6257 
6248 
6241 
6232 
6224 
6316 



LOG. COTAN. 

M 



98 



LOG. SINE. 



[Table ii. 



10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

40 
41 

42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



12° 

9-3176789 
84728 
90659 
96581 

9-3202495 
08400 
14297 
20186 
26066 
31938 

37802 
43657 
49505 
55344 
61174 
66997 
72811 
78617 
84416 
90206 



9-3301761 

07527 
13285 
19035 
24777 
30511 
36237 
41955 
47665 

53368 
59062 
64749 
70428 
76099 
81762 
87418 
93065 
98706 
9-3404338 

09963 
15580 
21190 
26792 
32386 
37973 
43552 
49124 
54688 
60245 
65794 
71336 
76870 
82397 
87917 
93429 
98934 
9-3504432 
09922 
15405 



770 



dif. 

59391 

5931 

5922 

5914 

590 

5897 

5889 

5872 
5864 

5855 
5848 
5839 
5830 
5823 
5814 
5806 
5799 
5790 
5782 

5773 
5766 

5758 
5750 
5742 
5734 
5726 
5718 
5710 
5703 
5694 
5687 
5679 
5671 
5663 
5656 
5647 
5641 
5632 
5625 

5617 
5610 
5602 
5594 
5587 
5579 
5572 
5564 
5557 
5549 
[5542 
5534 
5527 
55-20 
5512 
5505 
5498 
5490 
5483 
5475 

diff\ 



13° 

1-3520880 
26349 
31810 
37264 
42710 
48150 
53582 
59007 
64426 
69836 

75240 
80637 
86027 
91409 
96785 
•3602154 
07515 
12870 
18217 
23558 
28892 
34219 
39539 
44852 
50158 
55458 
60750 
66036 
71315 
76587 

81853 
87111 
92363 
97608 
•3702847 
08079 
13304 
18523 
23735 
28940 

34139 
39331 
44517 
49696 
54868 
60034 
65194 
70347 
75493 
80633 
85767 
90894 
96015 
1-3801129 
06237 
11339 
16434 
21523 
26605 
31682 
36752 
76° 



14° 

9-3836752 
41815 
46873 
51924 
56969 
62008 
67040 
72067 
77087 
82101 

87109 
92111 
97106 
3902096 
07079 
12057 
17028 
21993 
26952 
31905 

36852 
41794 
46729 
51658 
56581 
61499 
66410 
71315 
76215 
81109 

85996 
90878 
95754 
9-4000625 
05489 
10348 
15201 
20048 
24889 
29724 

34554 
39378 
44196 
49009 
53816 
58617 
63413 
68203 
72987 
77766 

82539 
87306 
92068 
96824 
9-4101575 
06320 
11059 
15793 
20522 
25245 
29962 
diff.l 750 



5469 
5461 
5454 
5446 
5440 
5432 
5425 
5419 
5410 
5404 

5397 
5390 
5382 
5376 
5369 
361 
5355 
5347 
5341 
5334 

5327 
5320 
5313 

5306 
5300 
5292 
5286 
5279 
5272 
5266 

5258 
252 
5245 
5239 
5232 
5225 
5219 
5212 
5205 
5199 
5192 
5186 
5179 
5172 
5166 
5160 
5153 
5146 
5140 
5134 

5127 
5121 
5114 
5108 
5102 
5095 
5089 
5082 
5077 
5070 



5063 
5058 
5051 
5045 
5039 
5032 
5027 
5020 
5014 
5008 
5002 
4995 
4990 
4983 
4978 
4971 
4965 
4959 
4953 
4947 

4942 
4935 
4929 
4923 
4918 
4911 
4905 
4900 
4894 
4887 
4882 
4876 
4871 
4864 
4859 
4853 
4847 
4841 
4835 
4830 
4824 
4818 
4813 
4807 
4801 
4796 
4790 
4784 
4779 
4773 

4767 
4762 
4756 
4751 
4745 
4739 
4734 
4729 
4723 
4717 



15° 

9-4129952 
34674 
39381 
44082 
48778 
52468 
58152 
62832 
67506 
72174 

76837 
81495 
86148 
90795 
95436 
9^4200073 
04704 
09330 
13950 
18566 
23176 
27780 
32380 
36974 
41563 
46147 
50726 
55299 
59867 
64430 



73541 
78089 
82631 
87169 
91701 
96228 
9^4300750 
05267 
09779 
14286 
18788 
23285 
27777 
32264 
36746 
41223 
45694 
50161 
54623 

59080 
63532 
67980 
72422 
76859 
81292 
85719 
90142 
94560 
I 98973 
19-4403381 
I 74° 



4712 
4707 
4701 
4696 
4690 
4684 
4680 
4674 
4668 
4663 

4658 
4653 
4647 
4641 
4637 
4631 
4626 
4620 
4616 
4610 
4604 
4600 
4594 
4589 
4584 
4579 
4573 
4568 
4563 
4558 

4553 
4548 
4542 
4538 
4532 
4527 
4522 
4517 
4512 
4507 
4502 
4497 
4492 
4487 
4482 
4477 
4471 
4467 
4462 
4457 
4452 
4448 
4442 
4437 
4433 
4427 
4423 
4418 
4413 
4408 



16 

1-4403381 
07784 
12182 
16576 
20965 
25349 
29728 
34103 
38472 
42837 
47197 
51553 
55904 
60250 
64591 
68927 
73259 
77586 
81909 
86227 

90540 
94849 
99153 
-4503452 
07747 
12037 
16322 
20603 
24879 
29151 

33418 
37681 
41939 
46192 
50441 
54686 
58926 
63121 
67392 
71618 

75840 
80058 
84271 
88480 
92684 
96884 
1-4601079 
05270 
09456 
13638 

17816 
21989 
26158 
30323 
34483 
38639 
42790 
46938 
51081 
55219 
59353 
730 



dif 

4403 

1398 
4394 
4389 
4384 
4379 
4375 
4369 
4365 
4360 

4356 
4351 
4346 
4341 
4336 
4332 
4327 
4323 
4318 
4313 

4309 
4304 
4299 
4295 
4290 
4285 
4281 
4276 
4272 
4267 
4263 
4258 
4253 
4249 
4245 
4240 
4235 
4231 
4226 
4222 

4218 
4213 
4209 
4204 
4200 
4195 
4191 
4186 
4182 
4178 
4173 
4169 
4165 
4160 
4156 
4151 
4148 
4143 
4138 
4134 

\diff. 



LOG. COSINE. 



Table ir.] 



LOG. TAN. 



99 



120 

9-3274745 
80953 
87153 
93345 
99528 

9-3305704 
11872 
18031 
24183 
30327 
36463 
42591 
48711 
54823 
6092' 
67024 
73113 
79194 
85267 
91333 

97391 
9-3403441 
09484 
15519 
21546 
27566 
33578 
39583 
45580 
51570 

57552 
63527 
69494 
75454 
81407 
87352 
93290 
99220 
9-3505143 
11059 
16968 
22869 
28763 
34650 
40530 
46402 
52267 
58126 
63977 
69821 

75658 
81487 
87310 
93126 
98935 
9-3604736 
10531 
16319 
22100 
27874 
33641 
77° 



diff. 

6208 
6200 
6192 
6183 
6176 
6168 
6159 
6152 
6144 
6136 

6128 
6120 
6112 
6104 
6097 
6089 
6081 
6073 
6066 
6058 
6050 
6043 
6035 
6027 
6020 
6012 
6005 
5997 
5990 
5982 

5975 
5967 
5960 
5953 
5945 
5938 
5930 
5923 
5916 
5909 
5901 
5894 
5887 
5880 
5872 
5865 
5859 
5851 
5844 
5837 

5829 
5823 
5816 
5809 
5801 
5795 
5788 
5781 
5774 
5767 

dii 



13° 

1.3633641 
39401 
45155 
50901 
56641 
62374 
68100 
73819 
79532 
85238 
90937 
96629 

1-3702315 
07994 
13667 
19333 
24992 
30645 
36291 
41930 
47563 
53190 
58810 
64423 
70030 
75631 
81225 
86813 
92394 
97969 

.3803537 
09100 
14655 
20205 
25748 
31285 
36816 
42340 
47858 
53370 

58876 
64376 
69869 
75356 
80837 
86312 
91781 
97244 
•3902700 
08151 

13595 
19034 
24466 
29893 
35313 
40727 
46136 
51538 
56935 
62326 
67711 
76° 



dil[ 

5760 
5754 
5746 
5740 
5733 
5726 
5719 
5713 
5706 
5699 
5692 
5686 
5679 
5673 
5666 
5659 
5653 
5646 
5639 
5633 
5S27 
5620 
5613 
5607 
5601 
5594 
4588 
5581 
5575 
5568 
5563 
5555 
5550 
5543 
5537 
5531 
5524 
5518 
5512 
5506 
5500 
5493 
5487 
5481 
5475 
5469 
5463 
5456 
5451 
5444 

5439 
5432 
5427 
5420 
5414 
5409 
5402 
5397 
5391 
5385 

diff. 



14° 

9-3967711 

73089 

78463 

83830 

89191 

94547 

99896 
9-4005240 

10578 

15910 

21237 

26558 

31873 

37182 

42486 

47784 

53076 

58363 

63644 

68919 

74189 
79453 
84712 
89965 
95212 
9-4100454 
05690 
10921 
16146 
21366 

26581 
31789 
36993 
42191 
47383 
52570 
57752 
62928 
68099 
73265 

78425 
83580 
68729 
93874 
99013 
9-4204146 
09275 
14398 
19515 
24628 

29735 
34838 
39935 
45026 
50113 
55194 
60271 
65342 
70408 
75469 
80525 
75° 

LOG. COTAN. 



diff 

5378 
5374 
5367 
5361 
5356 
5349 
5344 
5338 
5332 
5327 

5321 
5315 
5309 
5304 

5298 
5292 
5287 
5281 
5275 
5270 

5264 
5259 
5253 
5247 
5242 
5236 
5231 
5225 
5220 
5215 
5208 
5204 
5198 
5192 
5187 
5182 
5176 
5171 
5166 
5160 
5155 
5149 
5145 
5139 
5133 
5129 
5123 
5117 
5113 
5107 
5103 
5097 
5091 
5087 
5081 
5077 
5071 
5066 
5061 
5056 



95661 ^O^*- 



•4300697 
05727 
10753 
15773 
20789 
25799 
30804 
35805 
40800 
45791 
50776 

60733 
65704 
70670 
75631 
80687 
85538 
90485 
95426 
-4400363 
05295 
10222 
15145 
20062 
24975 

29883 
34786 
39685 
44579 
49468 
54352 
59232 
64107 
68978 
73843 



78704 
83561 
88413 
93260 
98102 
•4502940 
07774 
12602 
17427 
22246 
27061 
31872 
36678 
41479 
46276 
51069 
55857 
60641 
65420 
70194 
74964 
74° \diff. 



5036 
5030 
5026 
5020 
5016 
5010 
5005 
5001 
4995 
4991 
4985 
4981 
497G 
4971 
4966 
4961 
4956 

4951 
4947 
4941 
4937 
4932 
4927 
4923 
4917 
4913 
490S 

4903 
4899 
4894 
4889 
4884 
4880 
4375 
4871 
4865 
4861 
4857 
4852 
4847 
4842 
4838 
4834 
4828 
4825 
4819 
4815 



4811 
4806 
4801 
4797 
4793 
4788 
4784 
4779 
4774 
4770 



16° 

.4574964 
79730 
84491 
89248 
94001 
98749 

•4603492 
08232 
1296 
17697 

22423 
27145 
318e3 
36576 
41285 
45990 
50690 
55386 
60078 
64765 

69448 
74127 
78802 
83473 
88139 
92801 
97459 
-4702112 
06762 
11407 
16048 
20685 
25318 
29947 
34572 
39192 



48421 
53029 
57633 
62233 
66829 
71421 
76009 
80592 
85172 
89748 
94319 



•4803451 
08011 
12566 
17118 
21666 
26210 
30750 
35286 
39818 
44346 
48870 
53390 
73° 



diff. 

4766 
4761 
4757 
4753 

4748 
4743 
4740 
47351 
4730i 
4726 
4722 
4718 
4713 
4709 
4705 
4700 
4696 
4692 
4687 
4683 

4679 
4675 
4671 
4666 
4662 
4658 
4653 
4650 
4645 
4641 

4637 
4633 
4629 
4625 
4620 
4616 
4613 
4608 
4604 
4600 
4596 
4592 
4588 
4583 
4580 
4576 
4571 
4568 
4564 
4560 

4555 
4552 
4548 
4544 
4540 
4536 
4532 
4528 
4524 
4520 



300 



LOG. SINE. 



[Table n. 



10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

30 

31 

32 

33 

34 

.35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

59 

59 

60 



17° 
9-4659353 
63483 
67609 
71730 
75848 
79960 
84069 
88173 
92273 
96369 

9-4700461 
04548 
08631 
12710 
16785 
20356 
24922 
23985 
33043 
37097 

41146 
45192 
49234 
53271 
57304 
61334 
65359 
69380 
73396 
77409 



81418 
85423 
89423 
93420 
97412 
•4801401 
05385 
09366 
13342 
17315 
21283 
25248 
29208 
33165 
37117 
41066 
45010 
48951 
5288? 
56820 

60749 
64674 
68595 
72512 
76426 
80335 
84240 
88142 
92040 
95934 
99824 
72° 



diff. 

4130 

4126 
4121 

4116 
14112 
4109 
4104 
4100 
4096 
4092 

4037 
4083 
4079 
4075 
4071 
4066 
4063 
4058 
4054 
4049 
4046 
4042 
4037 
4033 
4030 
4025 
4021 
4016 
4013 
4009 
4005 
4000 
3997 
3992 



3984 
3981 
3976 
3973 
3968 

3965 
3960 
3957 
3952 
3949 
3944 
3941 
3937 
3932 
3929 

3925 
3921 
3917 
3914 
3909 
3905 
3902 
3396 
3894 
3390 

diff. 



18° 
•4899824 
•4903710 
07592 
11471 
15345 
19216 
23033 
26946 
30806 
34661 

38513 
42361 
46205 
50046 
53883 
57716 
61545 
65370 
69192 
73010 
76824 
80635 
84442 
83245 
92045 
95840 
99633 
■5003421 
07206 
10937 
14764 
18533 
22308 
26075 
29838 
33597 
37353 
41105 
44853 
48598 

52339 
56077 
59811 
63542 
67269 
70992 
74712 
78428 
82141 
85850 
89556 
93258 
96956 
1-5100651 
04343 
08031 
11716 
15397 
19074 
22749 
26419 
71° 



3886 

3882 

3379 

3374 

3371 

336 

3863 

3360 

3355 

3852 

3346 
3844 
3841 
3837 
3833 
3329 
38-25 
3322 
3315 
3314 

3311 
3807 
3803 
3300 
3795 
3793 
3786 
3785 
3781 
3777 

3774 
3770 
3767 
3763 
3759 
3756 
3752 
3748 
3745 
3741 

3738 
3734 
3731 
3727 
3723 
3720 
3716 
3713 
3709 
3706 

3702 
3698 
3695 
3692 
3688 
3635 
3681 
3677 
3675 
3670 



19° 

■5126419 
30086 
33750 
37410 
41067 
44721 
48371 
52017 
55660 
59300 
62936 
66569 
70196 
73624 
77447 
81066 
64682 
83-295 
91904 
95510 
99112 

■5202711 
06307 
09399 
13488 
17074 
20656 
24235 
27811 
31383 

34953 

38518 
42081 
45640 
49196 
52749 
56298 
59844 
63387 
6692 
70463 
73997 
77526 
81053 
84577 



91614 

95128 

98638 

^•5302146 

05650 
09151 
12649 
16143 
19635 
23123 
26608 
30090 
33569 
37044 
40517 
70° 

LOG. COSINE. 



3667 
3664 
36601 
3657 
3654 
3650 
3646 
3643 
3640 
3636 
3633 
3629 
3626 
3623 
3619 
3616 
3613 
3309 
3606 
3602 

3599 
3596 
3592 
3539 
3586 
3582 
3579 
3576 
3572 
3570 

3565 
3563 
3559 
3556 
3553 
3549 
3546 
3543 
3540 
3536 
3534 
3529 
3527 
3524 
35-20 
3517 
3514 
3510 
3508 
3504 

3501 
3496 
3494 
3492 
3486 
3465 
3462 
3479 
3475 
3473 



20° \diff. 

-5340517U.ftQ 

43966^2fifi 
47452^66 

543/004.- 
57832^^1^ 
61286'^]^i 
64737, ^^Ji 

68134^4' 
71624111 

'^P3438 

922301^1^3 
95653 2^20 

yyu 1 6 nj-, c 

■5402439:^^15 

05903,^115 

09314|" 

}^;^J:3405 

19527 ^^OJ 
22926 |99 

3392 

3390 

3336 

3384 

3380 

3377 

3375 

3371 

3369 

3365 

3362 

3360 

335' 

3353 

3351 

3347 

3345 

3342 

3339 

3336 

3333 

3330 

3327 

3324 

3321 

3319 
3315 
3313 
3310 
3307 
3304 
3301 
3298 
3295 
3293 



26321 
29713 
33103 
36489 
39873 

43253 
46630 
50005 
53376 
56745 
60110 
63472 
66832 
70189 
73542 

76893 
80240 
83585 
86927 
90266 
93602 
96935 
•5500265 
03592 
06916 
10237 
13556 
16871 
20184 
23494 
26801 
30105 
33406 
36704 
39999 
43292 



diff. 



21° 

•5543292 

46581 
49868 
53152 
56433 
59711 
6-2967 
66259 
69529 
72796 

76060 
79321 
8-2579 
85835 
89088 
92338 
95585 
96829 
•560-2071 
05310 

06546 

11779 

15010 

1823 

21462 

24685 

27904 

31121 

34335 

37546 

40754 

43960 

47163 

50363 

53561 

56756 

59948 

6313 

66324 

69508 

72689 
75868 
79044 
82217 
85387 
88555 
91721 
■ 94883 
98043 
1^5701200 

04355 
07506 
10656 
13802 
16946 
20087 
23226 
26362 
29495 
32626 
35754 



diff.i ' 

3289 f 2 
3287 59 
58 



3261 
3276 
3276 
3272 
3270 
3267 
3264 

3261 
3256 
3256 
3253 
3250 
3247 
3-244 
3242 
3239 
3236 
3233 
3231 
3227 
3225 
3223 
3219 
3217 
3214 
3211 
3208 

3206 
3203 
3200 
3193 
3195 
3192 
3189 
3187 
3184 
3181 

3179 
3176 
3173 
3170 
3168 
3166 
3162 
3160 
3157 
3155 
3151 
3150 
3146 
3144 
3141 
3139 
3136 
3133 
3131 
3128 

diff. 



LOG. TAN. 



101 i 



9-4S53390 
57907 
62419 
66928 
71433 
75933 
80430 
84924 
89413 
93898 



9-4902858 
07332 
11802 
16269 
20731 
25190 
29646 
34097 
38545 
42988 
47429 
51865 
56298 
60727 
65152 
69574 
73991 
78406 
82816 
87223 
91626 
96026 

9-5000422 
04814 
09203 
13588 
17969 
22347 
26721 

31092 
35459 
39822 
44182 
48538 
52891 
57-240 
61586 
65928 
70267 
74602 
78933 
83261 
87586 
91907 
96224 
9-5100539 
04849 
09156 
13460 
17760 
72o 



diff. 

4517 
4512 
4509 
4505 
4500 
4497 
4494 
4489 
4485 
4482 

4478 
4474 
4470 
4467 
4462 
4459 
4456 
4451 
4448 
4443 
4441 
4436 
4433 
4429 
4425 
4422 
4417 
4415 
4410 
4407 

4403 
4400 
4396 
4392 
4389 
4385 
4381 
4378 
4374 
4371 
4367 
4363 
4360 
4356 
4353 
4349 
4346 
4342 
4339 
4335 

4331 
4328 
4325 
4321 
4317 
4315 
4310 
4307 
4304 
4300 



18^ 
9-5117760 
22057 
26351 
30641 
34927 
39210 
43490 
47766 
52039 
56309 
60575 
64838 
69097 
73353 
77606 
81855 
86101 
90344 
94583 
98319 

9-5203052 
07282 
11508 
15730 
19950 
24166 
28379 
32589 
36795 
40999 
45199 
49395 
53589 
57779 
61966 
66150 
70331 
74508 
78682 
82853 
87021 
91186 
95347 
99505 

9-5303661 
07813 
11961 
16107 
20250 
24389 
28526 
32659 
36789 
40916 
45040 
49161 
53278 
57393 
61505 
65613 
69719 
710 



diff. 

4297 
4294 
4290 
4286 
4283 
4280 
4276 
4273 
4270 
4266 
4263 
4259 
4256 
4253 
4249 
4246 
4243 
4239 
4236 
4233 

4230 
4226 
4222 
4220 
4216 
4213 
4210 
4206 
4204 
4200 

4196 
4194 
4190 
4187 
4184 
4181 
4177 
4174 
4171 
4168 
4165 
4161 
4158 
4156 
4152 
4148 
4146 
4143 
4139 
4137 

4133 
4130 
4127 
4124 
4121 
4117 
4115 
4112 
4108 
4106 



19° 

9-5369719 
73821 
77920 
82017 
86110 
90200 
94287 
98371 

9-5402453 
06531 

10606 
14678 
18747 
22813 
26877 
30937 
34994 
39048 
43100 
47148 
51193 
55236 
59276 
63312 
67346 
71377 
75405 
79430 
83452 
87471 

91487 
95500 
99511 
9-5503519 
07523 
11525 
15524 
19521 
23514 
27504 
31492 
35477 
39459 
43438 
47415 
51388 
55359 
59327 
63292 
67255 
71214 
75171 
79125 
83077 
87025 
90971 
94914 



9-5602792 
06727 
10659 
70° 



dil 

4102 

4099 

4097 

4093 

4090 

4087 

4084 

4082 

4078 

4075 

4072 
4069 
4066 
4064 
4060 
4057 
4054 
4052 
4048 
4045 
4043 
4040 
4036 
4034 
4031 
4028 
4025 
4022 
4019 
4016 

4013 
4011 

4008 
4004 
4002 
3999 
3997 
3993 
3990 
3988 

3985 
3982 
3979 
3977 
3973 
3971 
3968 
3965 
3963 
3959 
3957 
3954 
3952 
3948 
3946 
3943 
3940 
3938 
3935 
3932 

diff. 



20° 
9-5610659 

14588 
18515 
22439 
26360 
30278 
34194 
38107 
42018 
45925 
49S31 
53733 
57633 
61530 
65424 
69316 
73205 
77091 
80975 
84856 

88735 
92611 
96484 
9-5700355 
04223 



11951 
15811 
19669 
23524 

27377 
31227 
85074 
38919 
42761 
46601 
50438 
54272 
58104 
61934 
65761 
69585 
73407 
77226 
81043 
84858 
88669 
92479 
96286 
19-5800090 
03892 
07691 
11488 
15282 
19074 
22864 
26651 
30435 
34217 
37997 
41774 
69° 



3929 
3927 
3924 
3921 
3918 
3916 
3913 
3911 
3907 
3906 
3902 
3900 
3897 
3894 
3892 
3889 
3886 
3884 
3881 
3879 

3876 
3873 
3871 
3868 
3865 
3863 
3860 
3858 
3855 
3853 
3850 
3847 
3845 
3842 
3840 
3837 
3834 
3832 
3830 
3827 



21° 

9-5841774 
45549 
49321 
53091 
56859 
60624 
64386 
08147 
71904 
75660 

79413 
83163 
86912 
90657 
94401 



3775 
3772 
3770 
3768 
3765 
3762 
3761 
3757 
[3756 
3753 

3750 
3749 
3745 
3744 



98142i?-~- 



3824 
3822 
3819 
3817 
3815 
3811 
3810 
3807 
3804 
3802 

3799 
3797 
3794 
3792 
3790 
3787 
3784 
3782 
3780 
3777 

diff. 



9-5901881 
05617 
09351 
13082 
16812 
20539 
24263 
27985 
31705 
35423 
39138 
42851 
46561 
50269 
53975 
57679 
61380 
65079 
68776 
72470 
76162 
79852 
83540 
87225 
90908 
94588 
98267 

9-6001943 
05617 
09289 
12958 
16625 
20290 
23953 

27613 
31271 

34927 
38581 
42233 
45882 
49529 
53174 
56817 
60457 
64096 
68° 



LOG. COTAN. 



3739 
3736 
3734 
3731 
3730 
3727 
3724 
3722 
3720 
3718 
3715 
3713 
3710 
3708 
3706 

3704 
3701 
3699 
3697 
3694 
3692 
3690 
3688 
3685 
3683 

3680 
3679 
3676 
3674 
3672 
3669 
3667 
3665 
3663 
3660 

3658 
3656 
3654 
3652 
3649 
3647 
3645 
3643 
3640 
3639 

diff. 



9* 



102 



LOG. SINE. 



[Table u. 





1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



22° 

9-5735754 
8880 

9-5742003 
5123 
8240 

9-5751356 
4468 
7578 

9-5760685 
3790 
6892 
9991 

9-5773088 
6183 
9275 

9-5782364 
5450 
8535 

9-5791616 
4695 
7772 

9-5800845 
3917 
6986 

9-5810052 
3116 
6177 
9236 

9-5822292 
5345 
8397 

9-5831445 
4491 
7535 

9-5840576 
3615 
6651 
9685 

9-5852716 
5745 

8771 
9-5861795 

4816 
7835. 

9-5870851 
3865 
6876 
9885 

9-5882892 
5896 

6897 

9-5891897 

4893 



3126 
3123 
312( 
3117 
3116 
3112 
3110 
3107 
3105 
3102 



9-5900880 
3869 
6856 
9841 

9-5912823 
5803 
8780 
67° 



3099 
3097 
3095 
3092 
3089 
3086 
3085 
3081 
3079 
3077 

3073 
3072 
3069 
3066 
3064 
3061 
3059 
3056 
3053 
3052 
3048 
3046 
3044 
3041 
3039 
3036 
3034 
3031 
3029 
3026 

3024 
3021 
3019 
3016 
3014 
3011 
3009! 
3007| 
3004! 
3001 
3000' 
2996: 
2995 
2992 
2989 
2987 
2985 
2982 
2980 
2977 



23° 

9-5918780 

9-5921755 
4728 
7698 

9-5930666 
3631 
6594 
9555 

9-594-2513 
5469 
8422 

9-5951373 
4322 
7268 
960212 
3154 
6093 
9030 

3-5971965 
4897 



diff. 



7827 

9-5980754 
367f: 
6602 
952: 

9-5992441 
5357 
8270 

9-6001181 
4090 
6997 
9901 

9-6012803 
5703 
8600 

9-6021495 
4388 
7278 

9-0030166 
3052 

5936 
8817 

9-6041696 
4573 
7446 

9-6050320 
3190 
6057 
8923 

:9-0061786 
4647 
7506 

9-6070362 
3216 
6068 
8918 

9-6081765 
4611 
7454 

9-6090294 

3133 

660 



diff. 

2975 

2973 

2970 

2968 

296i 

2963 

2961 

2958 

2956 

2953 

2951 

2949 

2946 

^944 

2942 

2939 

2937 

2935 

293k 

293 

2927 
•2925 
2923 
292] 
2916 
2916 
291c 
2911 
2909 
2907 
2904 
2902 
2900 
2897 
2895 
2893 
2890 
2838 



2884 

•2881 
2879 
2877 
2875 
2872 
2870 
2367 
2866 
2863 
2861 

■2859 
2856 
2354 
2852 
2850 
2847 
2846 
2843 
2840 
2839 

diff. 



24° \diff: 
9-6093133 

5969 

8803 
9-6101635 

4465 

7293 
9-6110118 

-2941 

5762 

8580 

9-6121397 

4211 

7023 

9833 
9-6132641 

5446 

8250 
9-6141051 

3650 

6647 

9441 
9-6152234 

5024 

7812 
9-6160599 

3382 

6164 

8944 
9-6171721 

4496 

7270 
9-6180041 

2809 

5576 

8341 
9-6191103 

3864 

6622 

9375 
9-6202132 



2836 
2834 
2832 
2830 
2828 
2825 
2823 
•2821 
2818 
>817 

2814 
2812 
281G 
2806 
2805 
2804 
2801 
2799 
•2797 
2794 

2793 

279C 
2788 
2787 
2783 
2782 
2780 
2777 
2775 
•2774 



7634 
9-6210382 

3127 

6871 

8612 
9-6221351 

4088 

6824 

9557 
9-6-232287 

5016 

7743 
9-6240468 

3190 

5911 

8629 
9-6251346 

4060 

6772 

9483 
6.5° 

LOG. COSINE. 



25° \diff.l 26° 
6259483 27Qg 9-6418420 

2706 

2704 

2702 

2700 

2698 

2696 

2693 

2692 

2690 

2688 

2685 

2684 

2682 

2679 

267! 

267 

2674 

2672 

2669 

2668 

2665 

2664 

•2661 

2660 

2657 

2656 

2653 

2652 

2650 



•2771 
2768 
2767 
2765 
2762 
2761 
2758 
•2756 
2754 
2752 

2750 
2748 
2745 
2744 
2741 
27-39 
2737 
2736 
2733 
2730 

2729 
2727 
2725 
2722 
2721 
•2718 
2717 
2714 
2712 
2711 



6262191 
4897 
7601 

9-6270303 
3003 
5701 
8397 

9-6281090 
3782 

6472 
9160 

9-6291845 
4529 
7211 
9890: 

9-6302568 
5243 
7917 

9-6310569 

3258 
5926 
8591 
19-6321255 
3916 
6576 
9233 

9-6331889 
4542 
7194 
9844 

9-6342491 
5137 
7780 

9-6350422 
3062 
5699 
8335 

9-6360969 
3601 
6231 
8859 

9-6371484 
4108 
6731 
9351 

9-6381969 
4585 
7199 
9812 

9-6392422 
5030 
7637 

9-6400241 
2844 
5445 
8044 

9-6410640 

3235 

5828 

8420 

64° 



9-6421009 
3596 
6182 
8765 

9-6431347 
3926 
6504 
9080 

9-6441654 

4226 
6796 
9365 

9-6451931 
4496 
7058 
9619 

9-6462178 
4735 
7290 
9844 

i9-6472395 
4945 
7492 

9-6480038 
2582 
5124 
7665 

9-6490203 
2740 



2647 
2646 
2643 
2642 
2640 
2637 
•2636 
2634 
2632 
2630 

2628 
2625 
2624 
2623 
2620 
2618 
2616 
2614 
2613 
2610 

2608 
2607 
2604 
2603 
2601 
2599 
2596 
2595 
2593 
2592 

diff. 



5274 
7807 
6500338 
2868 
5395 
7920 
19-6510444 
2966 
5486 
8004 



9-6520521 
3035 
5548 
8059 

9-6530568 
3075 
5581 
8084 

9-6540586 



5584 
8081 

6550575 
3068 
5559 
8048 

6560536 

3021 

5505 

7987 

9-6570468 

63° 



diff. 

2589 
2587 
2586 
2583 
2582 
2579 
2578 
2576 
2574 
2572 

2570 
2569 
2566 
2565 
2562 
2561 
2559 
2557 
2555 
2554 

2551 
2550 
2547 
2546 
2544 
2542 
2541 
2538 
2537 
2534 

2533 
2531 
2530 
2527 
2525 
2524 
2522 
2520 
2518 
2517 

2514 
2513 
2511 
2509 
2507 
2506 
2503 
2502 
2500 
2498 
2497 
2494 
2493 
2491 
2489 
2488 
2485 
2484 
2482 
2481 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 
9 
8 
7 
6 
5 
4 
3 
2 



diff. 



Table n.] 



LOG. TAN. 



103 



2-2^ 

9-6064096 
7732 

9-6071366 
4997 
8627 

9-6082254 
5880 
9503 

9-6093124 
6742 

9-6100359 
3973 
7586 

9-6111196 
4804 
8409 

9-6122013 
5615 
9214 

9-6132812 

6407 

9-6140000 
3591 
7180 

9-6150766 
4351 
7934 

9-6161514 
5093 
8669 

9-6172243 
5815 
9385 

9-6182953 
6519 

9-6190083 
3645 
7205 

9-6200762 
4318 
7872 

9-6211423 
4973 
8520 

9-6222066 
5509 
9150 

9-6232690 
6227 
9763 

9-6243296 
6827 

9-6250356 
3884 
7409 

9-6260932 
4454 
7973 

9-6271491 
5006 
8519 
67° 



3636 
3634 
3631 
3630 
3627 
3626 
3623 
3621 
3618 
3617 

3614 
3613 
3610 
3603 
3605 
3604 
3602 
3599 
3598 
3595 
3593 
3591 
3539 
3586 
3585 
3583 
3580 
3579 
3576 
3574 

3572 
3570 
3568 
3566 
3564 
3562 
3560 
3557 
3556 
3554 

3451 
3550 
3547 
3546 
3543 
3541 
3540 
3537 
3536 
3533 

3531 
3529 
3528 
3525 
3523 
3522 
3519 
3518 
3515 
3513 



23° 

9-627S519 

9-6-28-2031 
5540 
9048 

9-6292553 
6057 
9558 

9-6303058 
6556 

9-6310052 

3545 
7037 

9-6320527 
4015 
7501 

9-6330985 
4468 
7948 

9-6341426 
4903 
8378 

9-6351850 
5321 
8790 

9-6362257 
5722 
9185 

9-6372646 
6106 
9563 

9-6383019 
6473 
9925 

9-6393375 
6823 

9-6400269 
3714 
7156 

9-6410597 
4036 
7473 

9-6420908 
4342 
7773 

9-6431203 
4631 
8057 

9-6441481 
4903 
8324 

9-6451743 
5160 
8575 

9-6461988 
5400 
8810 

9-6472217 
5624 
9028 

9-6482431 
5831 



diff. 

3512 
3509 
3508 
3505 
3504 
3501 
3500 
3498 
3496 
3493 
3492 
3490 
3488 
3486 
3484 
3483 
3480 
3478 
3477 
3475 
3472 
3471 
3469 
3467 
3465 
3463 
3461 
3460 
3457 
3456 
3454 
3452 
3450 
3448 
3446 
3445 
3442 
3441 
3439 
3437 

3435 
3434 
3431 
3430 
3428 
3426 
3424 
3422 
3421 
3419 
3417 
3415 
3413 
3412 
3410 
3407 
3407 
3404 
3403 
3400 



24° 

9-6485831 
9230 

9-6492628 
6023 
9417 

9-650-2809 
6199 
9587 

9-6512974 
6359 
9742 

9-6523123 
6503 
9881 

9-6533-257 
6631 

9-6540004 
3375 
6744 

9-6550112 

3477 
6841 

9-6560204 
3564 
6923 

9-6570280 
3636 
6989 

9-6580341 
3692 
7041 

9-6590387 
3733 
7076 

9-6600418 
3758 
7097 

9-6610434 
3769 
7103 

9-6620434 
3765 
7093 

9-6630420 
3745 
7069 

9-6640391 
3711 
7030 

9-6650346 
3662 
6975 

9-6660288 
3598 
6907 

9-6670214 
3519 
6823 

9-6680126 
3426 
6725 
65° 



diff. 

3399 
3398 
3395 
3394 
3392 
3390 
3388 
3387 
3385 
3383 

3381 
3380 
3378 
3376 
3374 
3373 
3371 
3369 
3368 
3365 
3364 
3363 
3360 
3359 
3357 
3356 
3353 
3352 
3351 
3349 
3346 
3346 
3343 
3342 
3340 
3339 
3337 
3335 
3334 
3331 

3331 

3328 
3327 
3325 
3324 
3322 
3320 
3319 
3316 
3316 

3313 
3313 
3310 
3309 
3*307 
3305 
3304 
3303 
3300 
3299 



25° 

9-6686725 

9-6690023 
3319 
6613 
9906 

9-6703197 
6486 
9774 

9-6713060 
6345 

9628 

6722910 

6190 

9468 

9-6732745 
6020 
9294 

9-6742566 
5836 
9105 

9-6752372 
5638 
8903 

9-6762165 
5426 
8636 

9-6771644 
5201 
8456 

9-6781709 

4961 
8211 

9-6791460 
4708 
7953 

9-6801198 
4440 
7682 

9-6810921 
4160 

7396 

9-6820832 
3865 
7098 

9-6830328 
3557 
6785 

9-6840011 
3236 
6459 
9681 

9-6852901 
6120 
9338 

9-6882553 
5768 
8981 

9-6872192 
5402 
8611 

9-6881818 
64" 



3298 
3296 
3294 
3293 
3291 
3289 
3288 
3286 
3285 
3283 

3282 
3280 
3278 
3277 
3275 
3274 
3272 
3270 
3269 
3267 

3266 
3265 
3262 
3-261 
3260 
3258 
3257 
3255 
3253 
3252 

3250 
3249 
3248 
3245 
3245 
3242 
3242 
3239 
3239 
3236 

3236 
3233 
3233 
3230 
3229 
3228 
322G 
3225 
3223 
3222 

32-20 
3219 
3218 
3215 
3215 
3213 
3211 
3210 
3209 
3207 

diff. 



26° 

•6881818 
5023 
8227 

-6891430 
4631 
7831 

-6901030 
4226 
7422 

-6910616 

3809 
7000 

-6920189 
3378 
6565 
9750 

-6932934 
6117 
9298 

-6942478 

5656 
8833 

•6952009 
5183 
8355 

-69615-27 
4697 
7865 

-6971032 
4198 

7363 

-6980526 

3687 

6847 

-6990006 

3164 

6320 

9474 

•7002628 

5780 

8930 

' -7012080 
5227 
8374 

1-7021519 
4663 
7805 

1-7030946 
4086 
7225 

)-7040362 
3497 
6632 
9765 

1-7052897 
6027 
9156 

)-7052284 
5410 
8535 

)• 707 1659 
63° 



3205'^2 

3204 ^^ 

3203 

3201 

3200 

3199 

3196 

3196 

3194 

3193 

3191 

3189 

3189 

3187 

3185 

3184 

3183 

3181 

3180 

3178 

3177 
3176 
3174 
3172 
3172 
3170 
3168 
3167 
3166 
3165 
3163 
3161 
3160 
3159 
3158 
3156 
3154 
3154 
3152 
3150 

3150 
3147 
3147 
3145 
3144 
3142 
3141 
3140 
3139 
3137 

3135 
3135 
3133 
3132 
3130 
3129 
3128 
3126 
3125 
3124 

4iff 



LOG. COT AN. 



104 



[Table n. 



4 

5 

6 

7 

8 

9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
201 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



27 
9-6570468 

2946 
5423 
7898 

9-6580371 
2842 
5312 
7780 

9-6590246 
2710 

5173 
7633 

9-6600093 
2550 
5005 
7459 
9911 

9-6612361 
4810 
7257 
9702 

9-6622145 
4586 
7026 
9464 

9-6631900 
4335 
6768 
9199 

9-6641628 

4056 
6482 
8906 

9-6651329 
3749 
6168 
8586 

9-6661001 
3415 
5828 
8238 

9-6670647 
3054 
5459 
7863 

9-6680265 
2665 
5064 
7461 
9856 

9-6692250 
4642 
7032 
9420 

9-6701807 
4192 
6576 
8958 

9-6711338 
3716 
6093 
. 62° 



diff. 

2478 
2477 
2475 
2473 
2471 
2470 
2468 
2466 
2464 
2463 
2460 
2460 
2457 
2455 
2454 
2452 
2450 
2449 
2447 
2445 

2443 
2441 
2440 
2438 
243& 
2435 
2433 
2431 
2429 
242.8 

2426 
2424 
2423 
2420 
2419 
2418 
2415 
2414 
2413 
2410 

2409 
2407 
2405 
2404 
2402 
2400 
2399 
2397 
2395 
2394 



2392 
2390 

2388 
2387 
2385 
2384 
2382 
2380 
2378 
2377 



28° 

9-6716093 
8468 

9-6720841 
3213 
5583 
7952 

9-6730319 
2684 
5047 
7409 
9769 

9-6742128 
4485 
6840 
9194 

9-6751546 
3896 
6245 
8592 
6760937 
3281 
5623 
7963 
6770302 
2640 
4975 
7309 
9642 

9-6781972 
4301 

6629 
8955 

9-6791279 
3602 
5923 
8243 

9-6800560 
2877 
5191 
7504 

9816 

9-6812126 

4434 

6741 

9046 

9-6821349 

3651 

5952 

8250 

9-6830548 

2843 

5137 

7430 

9720 

6842010 

4297 

6583 



9-6851151 
3432 
5712 

i 61° 



diff. 

2375 
2373 
2372 
2370 
2369 
2367 
2365 
2363 
2362 
2360 

2359 
2357 
2355 
2354 
2352 
2350 
2349 
2347 
2345 
2344 

2342 
2340 
2339 
2338 
2335 
2334 
2333 
2330 
2329 
2328 

2326 
2324 
2323 
2321 
2320 
•2317 
2317 
2314 
2313 
2312 
2310 
2308 
2307 
2305 
2303 
2302 
2301 
2298 
2298 
2295 
2294 
2293 
2290 
2290 
2287 
2286 
2285 
2283 
2281 
2280 



9-6855712 
7991 

9-6860267 
2542 
4816 
7088 
9359 

9-68716-28 
3895 
6161 

8425 
9-6880688 
2949 
5209 
7467 
9723 
9-6891678 
4232 
6484 
8734 

6900983 
3231 
5476 
7721 
9964 

9-6912205 
4445 
6683 
8919 

9-6921155 
3388 
5620 
7851 

9-6930080 
2308 
4534 
6758 
8981 

9-6941203 
3423 
5642 
7859 

9-6950074 
2288 
4501 
6712 
8922 

9-6961130 
3336 
5541 

7745 
9947 

9-6972148 
4347 
6545 
8741 

9-6980936 
3129 
5321 
7511 
9700 
60° 



diff 

2279 
2276 
2275 
2274 
2272 
2271 
2269 
2267 
2266 
2264 

2263 
2261 
2260 
2258 
2256 
2255 
2254 
2252 
2250 
2249 
2248 
2245 
2245 
2243 
2241 
2240 
2238 
2236 
2236 
2233 
2232 
2231 
2229 
2228 
2226 
2224 
2223 
2222 
2220 
2219 
2217 
2215 
2214 
2213 
2211 
2210 
2208 
2206 
2205 
2204 
2202 
2201 
2199 
2198 
2196 
2195 
2193 
2192 
2190 
2189 

diff. 



30° 

9-6989700 
9-6991887 
4073 
6258 
8441 
9-7000622 
2802 
4981 
7158 
9334 

9-7011508 
3681 
5852 
8022 

9-7020190 
2357 
4523 
6687 
8849 

9-7031011 
3170 
5329 
7486 
9641 

9-7041795 
3947 
6099 
8248 

9-7050397 
2543 

4689 
6833 
8975 

9-7061116 
3256 
5394 
7531 
9667 

9-7071801 
3933 
6064 
8194 

9-7080323 
2450 
4575 
6699 
8822 

9-7090943 
3063 
5182 

7299 
9415 

9-7101529 
3642 
5753 
7863 
9972 

9-7112080 

4186 

6290 

8393 

59° 



2187 
2186 
2185 
2183 
2181 
2180 
2179 
2177 
2176 
2174 

2173 
2171 
2170 
2168 
2167 
2166 
2164 
2162 
2162 
2159 

2159 
2157 
2155 
2154 
2152 
2152 
2149 
2149 
2146 
2146 
2144 
2142 
2141 
2140 
2138 
2137 
2136 
2134 
2132 
2131 

2130 
3129 
2127 
2125 
2124 
2123 
2121 
2120 
2119 
2117 

-2116 
2114 
2113 
2111 
2110 
2109 
2108 
2106 
2104 
2103 



31° 

9-7118393 

9-7120495 
2596 
4695 
6792 
8889 

9-7130983 
3077 
5169 
7260 
9349 

9-7141437 
3524 
5609 
7693 
9776 

9-7151857 
3937 
6015 
8092 

9-7160168 
2243 
4316 
6387 
8458 

9-7170526 
2594 
4660 
6725 
8789 

9-7180851 
2912 
4971 
7030 
9086 

9-7191142 
3196 
5249 
7300 
9350 

9-7201399 
3447 
5493 
7538 
9581 

9-7211623 
3664 
5704 
7742 
9779 

9-7221814 



5881 
7913 
9943 

9-7231972 
4000 
6026 
8051 

9-7240075 
2097 
58° 



2102 
2101 
2099 
2097 
2097 
2094 
2094 
2092 
2091 
2089 
2088 
2087 
2085 
2084 
2083 
2081 
2080 
2078 
2077 
2076 

2075 
2073 
2071 
2071 
2068 
2068 
2066 
2065 
2064 
2062 

2061 
2059 
2059 
2056 
2056 
2054 
2053 
2051 
2050 
2049 

2048 
2046 
2045 
2043 
2042 
2041 
2040 
2038 
2037 
2035 
2034 
2033 
2032 
2030 
2029 
2028 
2026 
2025 
2024 
2022 

diff. 



LOG. COSINE. 



Table ii.] 



LOG. TAN. 



]05 



09-7071659 



4781 
7902 

9-70S1022 
4141 
7258 

9-7090374 
3488 
6601 
9713 

9-7102824 
5933 
9041 

9-7112148 
5254 
8358 

9-7121461 
4562 
7662 

9-7130761 

3859 
6956 

9-7140051 
3145 
6237 
9329 

9-7152419 
5508 
8595 

9-7161682 

4767 
7851 

9-7170933 
4014 
7094 

9-7180173 
3251 
6327 
9402 

9-7192476 

5549 
8620 

9-7201690 
4759 
7827 

9-7210893 
3958 
7022 

9-7220085 
3147 

6207 
9266 

9-7232324 
5381 
8436 

9-7241490 
4543 
7595 



9-7250646 
3695 
6744 
620 



diff. 

3122 
3121 
3120 
3119 
3117 
3116 
3114 
3113 
3112 
3111 

3109 
3108 
3107 
3106 
3104 
3103 
3101 
3100 
3099 
3098 

3097 
3095 
3094 
3092 
3092 
3090 
3089 
3087 
3087 
3085 



28° 

9-7256744 
9791 

9-7262837 
5331 
8925 

9-7271967 
5008 
8048 

9-7281087 
4124 

7161 
9-7290196 
3230 
6263 
9295 
9-7302325 
5354 



3084 
3082 
3081 
3080 
3079 
3078 
3076 
3075 
3074 
3073 
3071 
3070 
3069 
3068 
3066 
3065 
3064 
3063 
3062 
3060 

3059 
3058 
3057 
3055 
3054 
3053 
3052 
3051 
3049 
3049 



9-7311410 
4436 

7460 

9-7320484 

3506 

6527 

9547 

9-7332566 

5584 

8601 

9-7341616 

4631 

7644 
9-7350656 
3667 
6677 
9685 
9-7362693 
5699 
8705 
9-7371709 
4712 

7714 
9-7380715 
3714 
6713 
9710 
9-7392707 
5702 
8696 
9-7401689 
4681 

7672 
9-7410662 
3650 
6638 
9624 

7422609 
5594 
8577 

7431559 
4540 
7520 
Clo 



304' 
3046 
3044 
3044 
3042 
3041 
3040 
3039 
3037 
3037 

3035 
3034 
3033 
3032 
3030 
3029 
3029 
3027 
3026 
3024 

3024 
3022 
3021 
3020 
3019 
3018 
3017 
3015 
3015 
3013 
3012 
3011 
3010 
3008 
3008 
3006 
3006 
3004 
3003 
3002 

3001 
2999 
2999 
2997 
2997 
2995 
994 
2993 
2992 
2991 



990 

2988 
2988 
2986 
2985 
2985 
•2983 
2982 
2981 
2980 

diff. 



29° 

9-7437520 

97440499 

3476 

6453 

9428 

9-7452403 

5376 

8349 

9-7461320 

4290 

7259 

9-7470227 

3194 

6160 

9125 

9-7482089 

5052 

8013 

9-7490974 

3934 

6892 



9-7502806 
5762 
8716 
7511669 
4622 
7573 

9-7520523 
3472 

6420 
9368 

9-7532314 
5259 
8203 

9-7541146 
4088 
7029 
9969 

9-7552908 

5846 
8783 

9-7561718 
4653 
7587 

9-7570520 
3452 
6383 
9313 

9-7582242 

5170 
8096 

9-7591022 
3947 
6871 
9794 

9-7602716 
5637 
8557 

9-7611476 
4394 
60° 



2979 
2977 
2977 
2975 
2975 
2973 
2973 
2971 
2970 
2969 
2968 
2967 
2966 
2965 
2964 
2963 
2961 
2961 
2960 
2958 



2958 
2956 
2956 
2954 
2953 
2953 
2951 
2950 
2949 
2948 

2948 
2946 
2945 
2944 
2943 
2942 
2941 
2940 
2939 
2938 
2937 
2935 
2935 
2934 
2933 
2932 
2931 
2930 
2929 
2928 

2926 
2926 
2925 
2924 
2923 
2922 
2921 
2920 
2919 
2918 

diff 



30° 

9-7614394 
7311 

9-7620227 
3142 
6056 
8969 

9-7631881 
4792 
7702 

9-7640612 

3520 
6427 
9334 

9-7652239 
5143 
8047 

9-7660949 
3851 
6751 
9651 

9-7672550 
5448 
8344 

9-7681240 
4135 
7029 
9922 
769-2814 
5705 
8596 

9-7701485 
4373 
7261 

9-7710147 
3033 
5917 
8801 

9-7721684 
4566 
7447 

9-7730327 
3206 
6084 
8961 

9-7741838 
4713 
7588 

9-7750462 
3334 
6206 

9077 
9-7761947 
4816 
7685 
9-7770552 
3418 
6284 
9149 
9-7782012 
4875 
7737 
59° 



diff. 

2917 
2916 
2915 
2914 
2913 
2912 
2911 
2910 
2910 
2908 
2907 
2907 
2905 
2904 
2904 
2902 
2902 
2900 
2900 
2899 



2896 
2896 
2895 
2894 
2893 
2892 
2891 
2891 



2888 



2884 
2884 
2883 
2882 
2881 
2880 

2879 

2878 
2877 
2877 
2875 
2875 
2874 
2872 
2872 
2871 
2870 



2867 
2866 
2866 
2865 
2863 
2863 
2862 

diff 



31° 

9-7787737 

9-7790599 
3459 
6318 
9177 

9-7802034 
4891 
7747 

9-7810602 
3456 

6309 
9162 

9-7822013 
4864 
7713 

9-7830562 
3410 
6258 
9104 

9-7841949 
4794 
7638 

9-7850481 
3323 
6164 
9004 

9-7861844 
4682 
7520 

9-7870357 

3193 
6028 
8863 

9-7881696 
4529 
7361 

9-7890192 
3023 
5852 
8681 

9-7901508 
4335 
7161 
9987 

9-7912811 
5635 
8458 

9-7921280 
4101 
6921 

9741 
9-7932560 
5378 
8195 
9-7941011 
3827 
6641 
9455 
9-7952268 
5081 
7892 
58° 



diff. 

2862 
2860 
2859 
2859 
2857 
2857 
2856 
2855 
2854 
2853 

2853 
2851 
2851 
2849 
2849 
2848 
2848 
2846 
2845 
2845 
2844 
2843 
2842 
2841 
2840 
2840 
2838 
2838 
2837 
2836 

2835 
2835 
2833 
2833 
2832 
2831 
2831 
2829 
2829 
2827 

2827 
2826 
2826 
2824 
2824 
2823 
2822 
2821 
2820 
2820 

2819 
2818 
2817 
2816 
2816 
2814 
2814 
2813 
2813 
2811 

diff. 



LOG. COTAN. 



106 



LOG. SINE. 



[Table n. 





1 

2 

3 

4 

5 

6 

7 

8 

9 
10 
111 

!12 

13 
14 
!l5 

:i6 

17 
18 
19 

20 
21 

|i 

24 
i25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



32^ \diff. 

9-7242097Ln^, 

4116o 



4204|2013 

02U 19019 

8229;X 7 
9-7260240.20J1 

?~f5!200S 

D2b4 V,r,n;, 

8269l;gg^ 
9-7270273|;gg43 

f.:;-^ 1 2002 
6-278 2000 



8277 
9-7280275 
2271 
4267 
6260 
8253 



613 

8156 

r4 

2189 



020 

2013 

iiV^2015 



33^ Idiff. 

9-7361083Lq44 
3032 1^5^ 

9-7370799;}^oc 

i'it 193S 
4b /o 



1999 
1998 
1996 
1996 
1993 
1993 



9-^290244j;|J 

6211 '^938 

8197! J^l 

9-73001821 J^g 



21651 

4148 



1983 
1931 



8109, 9^0 
9-7310087! I^i? 

4040}^."' 
6015 {^.4 
7989 \l'^ 



9-7321932i 
3902 



5870 
7837 
9803 
9-7331763 
3731 



34° \diff. 
9-74756171 ^g^2 



9-7481230 
3099 
4967 



1936 
1935 
1933 



6611 
6546 

9-7380479,000 
2419 iyo6 

'1931 
8201 ^9^- 



669S' 
9-7490562: 



9-73901-29 
2055 
3930 
5904 
7827 



1925 
19-26 
19-25 
19-24 
19-23 
1921 



914U! 
9-7510991 



9-740166S-J92P 
=n;i915 



550c 
7421 
9337 
9-74ir251 



11916 
!1916 
1914 

50?f^9n 

fi2Rfi^l911 

69B6^^gQg 



8895; 

9-7420303 
2710 
4616 
65-20 

9-74303-25' 9U^ 



1903 
1907 
1906 
1904 
1303 



2226 
41-26 
6024 



5693 
7654 
9614 

9-7341572' 
3529 
5435; 
7440 
9393 

9-7351345 
3296 



1970 
1968 
1967 
1966 
1965 
1963 
1962 
1961 
1960 
11953 
1957 
1956 
1955 
1953 
1952 
1951 



r9^f;;1950 

7195 ^9^-^ 



9142 
9-7361088 
57^ 



194 
1946 

diff. 



35 

9-7585913 



7489! 



1371 



misio 



1369 
1368 



6333. c, 



1366 



1365 
1364 



2425; 5, 



1363 



423 
6143 



1362 
i.il361 



771?!1804 

95'9'1302 

9-759132l!l|02 

b/lb^-g^ 

8515|,7q; 

600311 jlq^ 

2106 |19^ 



8007 



^,1359 



1359 



9366 ^p^7 

•7501723 qB' 

3579 Y41 

^434JS53 



1901 
1900 
1393 
1397 

9317 1396 
9-7441712 Jg, 
3606 -, gn^ 

7390 ^9 

9230 }x^^ 
9-7451169 IS^- 

3056 \°^L 

4943 jHS: 

6323 

8712 
9-7460595 

2477 

4353 

623' 

811 

999,,^-. 
9-7471863 |5i° 

0740 15/ o 

"^'^ ;1874 



1853 
1351 
jl351 
2842, iQ4q 

6^^H 1347 
8335 1Q4^ 

-75-20-231 ^^ 
2075^ g4 

3919, 049 
5761: ^4- 

7602 g^A 

-7531280 iqoQ 
3113:}„fi 

6'90;ipo4 

'•7540457! iqoi 

4jSiij 

E?49 132; 

■" 1I32: 



3399 
5692 
7433 
9274 
9-7611063 
2351 
4633 
6424 
8208 
9992 

9-7621775 
3556 
533 



96041 

9-7551431 

3-256 

5030 



1334 
1533 

1382 
1331 

1377 



5617| 
56^ 



1827 
1S25 
13-24 

1 P99 

6902; jp;:; 

8724'1^22 

"7560544 -,c9n 
2364: g^^ 
4132-5-^- 



5999 



\diff. 



36^ 

9-7692187 
3925 
5662 
7398 
9134 

9-7700866 
2601 



1793 
1791 
1791 
1789 
1783 
1737 
1786 
1734 
1734 
1733 
1731 
1781 

/116l}n9 
6394 1 J ^7 
9-763067L;ii6 

2^^1775 



5996 
7769 
9540 



1774 
1773 
1771 
1771 



9-764131L.-^Q 

4349 H^7 

6616 1'°' 



jsr 

_ ^1816 

7315 , 01 - 

9630. |5t^ 

■7571444! § 4 

3-256! -; 

i|l3}5 
8637;}S09 

7o8049o!inn'' 



2302 



4103 
5913 



cl306 



1305 
\diff. 



8382 
9-7650147 
1911 
3674 
5436 
7197 
8957 
9-7660715 
2473 
4229 
5985 
7739 
9492 
9-7671244 
2996 

4746 
6494 
6-242 
9939 

9-7681735 
3430 
52-23 
6966 
870' 

9-769044; 
218 
54= 



1766 

1765 

1764 

1763 

1762 

1761 

1760 

1753 

175 

1756 

1756 

1754 

1753 

1752 

1752 

1750 

1748 

1748 

1747 

1746 

1745 

1743 

1743 

1441 

1741 

1739 

diff. 



diff. 

1738 
1737 
1736 
1736 
1734 
1733 
1731 



4332L 

6063!t-nn 
77QQll'30 

"^f 17-29 
9522 



9-7711-249 
2976 
4702 
6426 
8150 
9372 

9-7721593 
3314 
5033 



67f 11717 



1727 
1727 
17-26 
1724 
1724 
1722 
172r 
1721 
1719 
1718 



S463>: :!39i 
9-7730135 |i;:i33 
1900;}i.|^{37 
3614i|'t^36 
53271 1 .^135 
7039'i-|pi34 
6749 ^l^ In' 33 

l/10|or,| 

1709j^Ti 
1708 "^^! 
3376 



9-7740459 
2168 



5583 
7283 
8993 
9-7750697 
2399 



1707 
11705 
1705 
1704 
1702 
4101 1'02 



5301! 155 -23; 

7501 22 

qi 00:1698 9, 

9^^^!1693j ^ 
^■''^§l?5il696 20 

2o93U(:.Q^:19 

42S9i|2Q5il8 
59S3!}694!i7 

7676'i/.nQ'16| 
9369 }S? 15' 

/i/ou 1 /:aq io 
4439,}6l9l2 



6123 

7815 



9501 
9-7781186 



2570 



1687 

1686 

1635 
7nl6S4 



4553 1683 
fi9Q^il682 

fqfg'iesi 

9596 }^30 
9-7791275 |6^9 

46301167^1 
53^ \diff\ 



LOG. COSINE. 



Table ii. 



LOG. TAN. 



107 



10 
11 

12 
13 
14 
15 
16 
17 
18 
19 
20 
121 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 



3-2° 

9-7957892 

9-7960703 
3513 
6322 
9130 

9-7971938 
4745 
7551 

9-7980356 
3160 



5964 
8767 

9-7991569 
4370 
7170 
9970 

9-8002769 
5567 
8365 

9-8011161 
3957 
6752 
9546 

9-8022340 
5133 
7925 

9-8030716 
3506 
6296 
9085 

9-8041873 
4661 
7447 

9-8050233 
3019 
5803 
8587 

9-8061370 
4152 
6933 
9714 

9-8072494 
5273 
6052 



diff. 

2811 

2810 
2809 



3606 
6383 
9158 
•8091933 
4707 

7480 
9-8100253 
3025 
5796 
8566 
9-8111336 
4105 
6873 
9641 
9-8122408 
5174 
57° 



2807 
2806 
2805 
2804 
2804 

2803 
2802 
2801 
2800 
2800 
2799 
2798 
2798 
2796 
2796 

2795 
2794 
2794 
2793 
2792 
2791 
2790 
2790 
2789 
2788 

2788 
2786 
2786 
2786 
2784 
2784 
2783 
2782 
2781 
2781 
2780 
2779 
2779 
2777 
2777 
2777 
2775 
2775 
2774 
2773 

2773 
2772 
2771 
2770 
2770 
2769 
2768 
2768 
2767 
2766 

diff. 



33° \diff. 
9-8125174 

7939 
9-8130704 

3468 

6231 

8993 
9-8141755 

4516 

7277 
9-8150036 

2795 

5554 

8311 
9'8161068 

3824 

6580 

9335 
9-8172089 

4842 

7595 



9-8180347 
3098 
5849 
8599 

9-8191348 
4096 
6844 
9592 

9-8202338 
5084 

7829 

9-8210574 
3317 
6060 
8803 

9-8221545 
4286 
7026 
9766 

9-8232505 
5244 
7981 

9-8240719 
3455 
6191 
8926 

9-8251660 
4394 
7127 
9860 

9-8262592 
5323 
8053 
8270783 
3513 
6241 
8969 

9-8281696 

4423 

7149 

9874 

560 



12765 
1 2765 
1 2764 
2763 
2762 
2762 
2761 
2761 
2759 
2759 

2759 
2757 
2757 
2756 
2756 
2755 
2754 
2753 
2753 
2752 

2751 
2751 
2750 
2749 

2748 
2748 
2748 
2746 
2746 
2745 

2745 
2743 
2743 
2743 
2742 
2741 
2740 
2740 
2739 
2739 
2737 
2738 
2736 
2736 
2735 
2734 
2734 
2733 
2733 
2732 
2731 
2730 
2730 
2730 
2728 
2728 
2727 
2727 
2726 
2725 



diff. 



34° 

9-8289874 

9-8292599 
5323 
8047 

9-8300769 
3492 
6213 
8934 

9-8311654 
4374 



7093 

9811 
9-8322529 

5246 

7963 
9-8330679 

3394 

6109 

8823 
9-8341536 

4249 

6961 

9673 
9-8352384 

5094 

7804 
9-8360513 

3221 

5929 

8636 
9-8371343 

4049 

6755 

9460 
9-8382164 

4867 

7571 
9-8390273 

2975 

5676 

8377 
9 8401077 

3776 

6475 

9174 
9-8411871 

4569 

7265 

9961 
9-8422657 

5351 

6046 
9-8430739 

3432 

6125 

8817 
9-8441508 

4199 

6889 

9579 
9-8452-268 

LOG. COT AN. 



diff. 

2725 
2724 
2724 
2722 
2723 
2721 
2721 
2720 
■2720 
2719 

2718 
■2718 
•2717 
2717 
2716 
2715 
2715 
•2714 
•2713 
'2713 
2712 
2712 
2711 
2710 
2710 
2709 
2708 
2708 
2707 
2707 

2706 
2706 
2705 
2704 
2703 
2704 
2702 
2702 
'2701 
•2701 
2700 
2699 
2699 
2699 
2697 
2698 
2696 
2696 
2696 
2694 

2695 
2693 
2693 
2693 
2692 
•2691 
•2691 
2690 
2690 
2689 

diff 



350 

9-8452268 
4956 
7044 

9-8460332 
3018 
5705 
8390 

9-8471075 
3760 
6444 

9127 

9-8481810 
4492 
1\1A 
9855 

3-8492536 
5216 
7S96 

9-8500575 
3253 
5931 
8608 

9-8511285 
3961 
6637 
9312 

9-8521987 
4661 
7335 

9-853000f 

2680 
5352 
8023 

9-8540694 
3365 
6034 
8704 

3-8551372 
4041 
6708 

9376 

9-8562042 

4708 

7374 

9-8570039 

2704 

5368 

8031 

8580694 

3357 

6019 
8680 

9-8591341 
4002 
6661 
9321 

9-8601980 
4638 
7296 
9954 

9-8612610 
54° 



diff 

'2688 

2G88 

2688 

2686 

2687 

2685 

2685 

2685 

2684 

2683 

2683 

2682 

2682 

2681 

2681 

2680 

2680 

2679 

2678 

2678 

2677 

2677 

2676 

2676 

2675 

2675 

267 

2G74 

2673 

2672 

'267 

2671 

2671 

2671 

2669 

2670 

2668 

2669 

2667 

2668 

2666 

2666 

2666 

2665 

2665 

2664 

2663 

2663 

2663 

2662 

2661 
2661 
2661 
2659 
2660 
2659 
2658 
2658 
2658 
2656 

diff. 



360 

9-8612610 
5267 
7923 

9-8620578 
3233 
5887 
8541 

9-8631195 
3848 
6500 
9152 

9-8641803 
4454 
7105 
9755 

9-8652404 
5053 
7702 

9-8660350 
2997 
5644 
8'291 

9-8670937 
3583 
6228 
8873 

9-8681517 
4160 
6804 
9446 

9-869 
4731 
7372 

9-8700013 
2653 
5293 
7933 

9-8710572 
3210 
5848 

8486 

9-8721123 
3760 
6396 
9032 

9-8731668 
4302 
6937 
9571 

9-8742204 
4838 
7470 

9-8750102 
2734 
5365 
7996 

9-8760627 
3257 



8515 
9-8771144 
53° 



2657 
2656 
2655 
2655 
2654 
2654 
2654 
2653 
2652 
•2652 

2651 
2651 
•2651 
2650 
2649 
2649 
2649 
2648 
2647 
2647 

'2647 
2646 
2646 
2645 
2645 
2644 
2643 
2644 
2642 
2643 

2642 
•2641 
2641 
2640 
2640 
2640 
2639 
2638 
2638 
•2638 
2637 
2637 
2636 
2636 
2636 
2634 
2635 
2634 
2633 
2634 

2632 
2632 
2632 
2631 
2631 
2631 
2630 
2629 
2629 
2629 



1108 



LOG. SINE. 



[Table n. 



37° 

9-7794630 
6306 
7981 
9655 

9-7801328 
3000 
4671 
6341 
8010 
9677 

9-7811344 
3010 
4675 
6339 
8002 
9664 

9-7821324 
2984 
4643 
6301 
7958 
9614 
7831268 
2922 
4575 
6227 
7878 
9528 

9-7841177 
2824 

4471 
6117 
7762 
9406 

9-7851049 
2691 
4332 
5972 
7611 
9249 

9-7860886 
2522 
4157 
5791 
7424 
9056 

9-7870687 
2317 
3946 
5574 

7202 
8828 

9-7880453 
2077 
3701 
5323 
6944 
8565 

9-7890184 
1802 
3420 
52' 



dijj 

1676 
1675 
1674 
1673 
1672 
1671 
1670 
1669 
1667 
1667 

1666 
1665 
1664 
1663 
1662 
1660 
1660 
1659 
1658 
1657 

1656 
1654 
1654 
1653 
1652 
1651 
1650 
1649 
1647 
1647 

1646 
1645 
1644 
1643 
1642 
1641 
1640 
1639 
1638 
1637 

1636 
1635 
1634 
1633 
1632 
1631 
1630 
1629 
1628 
1628 
1626 
1625 
1624 
1624 
1622 
1621 
1621 
1619 
1618 
1618 



38^ 

9-7893420 
5036 
6652 
8266 
9880 

9-7901493 
3104 
4715 
6325 
7933 
9541 

9-7911148 
2754 
4359 
5963 
7566 
9168 

9-7920769 
2369 
3968 



dil 



5566 
7163 
8760 

9-7930355 
1949 
3543 
5135 
6727 
8317 
9907 

9-7941496 
3083 
4670 
6256 
7841 
9425 

9-7951008 
2590 
4171 
5751 
7330 
8909 

9-7960486 
2062 
3638 
52i2 
6786 
8359 
9930 

9-7971501 

3071 
4640 
6208 
7775 
9341 
9-7980906 
2470 
4034 
5596 
7158 
8718 
5P 



1616 
1616 
1614 
1614 
1613 
1611 
1611 
1610 
1608 
1608 
1607 
1606 
1605 
1604 
1603 
1602 
1601 
1600 
1599 
1598 
1597 
1597 
1595 
1594 
1594 
1592 
1592 
1590 
1590 
1589 

1587 
1587 
1586 
1586 
1584 
1583 
1582 
1581 
1580 
1579 



1579 
1577 
1576 
1576 
1574 
1574 
1573 
1571 
1571 
1570 

1569 
1568 
1567 
1566 
1565 
1564 
1564 
1562 
1562 
1560 

diff. 



39° 

9-7988718 
9-7990278 
1836 
3394 
4951 
6507 
8062 
9616 
8001169 
2721 
4272 
5823 
7372 
8921 
9-8010468 
2015 
3561 
5106 
6649 
8192 

9735 
9-8021276 
2816 
4355 
5894 
7431 
8968 
9-8030504 
2038 
3572 

5105 
6637 

8168 
9699 
9-8041228 
2757 
4284 
5811 
7336 
8861 

9-8050385 
1908 
3430 
4951 
6472 
7991 
9510 

9-8061027 
2544 
4060 
5575 
7089 
8602 

9-8070114 
1626 
3136 
4646 
6154 
7662 
9169 

9-8080675 
50° 



1560 
1558 
1558 
1557 
1556 
1555 
1554 
1553 
1552 
1551 
1551 
1549 
1549 
1547 
1547 
1546 
1545 
1543 
1543 
1543 
1541 
1540 
1539 
1539 
1537 
1537 
1536 
1534 
1534 
1533 
1532 
1531 
1531 
1529 
1529 
1527 
15-2'7 
1525 
1525 
1524 

1523 
1522 
1521 
1521 
1519 
1519 
1517 
1517 
1516 
1515 
1514 
1513 
1512 
1512 
1510 
1510 
1508 
1508 
1507 
1506 

diff. 



40° 

9-8080675 
2180 
3684 
5188 
6690 
8192 
9692 

9-8091192 
2691 
4189 

568.6 

7182 



9'8100172 
1666 
3159 
4650 
6141 
7631 
9121 

9-8110609 
2096 
3583 
5069 
6554 
8038 
9521 

9-8121003 
2484 
3965 
5444 
6923 
8401 
9878 

9-8131354 
2829 
4303 
5777 
7250 
8721 

9-8140192 
1662 
3131 
4600 
6067 
7534 
8999 

9-8150464 
1928 



3391 
4854 
63,15 
7776 
9235 
9-8160694 
2152 
3609 
5066 
6521 
7975 
9429 
49° 



diff. 

1505 
1504 
1504 
1502 
1502 
1500 
1500 
1499 
1498 
1497 

1496 
1496 
1494 
1494 
1493 
1491 
1491 
1490 
1490 
1488 
1487 
1487 
1486 
1485 
1484 
1483 
1482 
1481 
1481 
1479 

1479 
1478 
1477 
1476 
1475 
1474 
1474 
1473 
1471 
1471 

1470 
1469 
1469 
1467 
1467 
1465 
1465 
1464 
1463 
1463 



410 

9-8169429 
9-8170882 
2334 
3785 
5235 
6685 
8133 
9581 
9-8181028 
2474 

3919 
5364 
6807 
8250 
9692 
9-8191133 
2573 
4012 
5450 



1461 
1461 
1459 
1459 
1458 
1457 
1457 
1455 
1454 
1454 



8325 
9761 

9-8201196 
2630 
40'63 
5496 
6927 
8358 
9788 

9-8211217 

2646 
4073 
5500 
6926 
8351 
9775 

9-8221198 
2621 
4042 
5463 
6883 
8302 
9721 

9-8231138 
2555 
3971 
5386 
6800 
8213 
9626 

9-8241037 
2448 
3858 
5267 
6676 
8083 
9490 

9-8250896 

2301 

3705 

5109 

48° 



1453 
1452 
1451 
1450 
1450 
1448 
1448 
1447 
1446 
1445 

1445 
1443 
1443 
1442 
1441 
1440 
1439 
1438 
1438 
1437 
1436 
1435 
1434 
1433 
1433 
1431 
1431 
1430 
1429 
1429 
1427 
1427 
1426 
1425 
1424 
1423 
1423 
1421 
1421 
1420 



1419 
1419 
1417 
1417 
1416 
1415 
1414 
1413 
1413 
1411 

1411 
1410 
1409 
1409 
1407 
1407 
1406 
1405 
1404 
1404 

dit 



58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
U 
10 



LOG. COSINE. 



Table II.] 



LOG. TAN. 



109 



37° 

9-8 71144 
3772 
6400 
9027 

9-8781654 
428] 
6907 
9533 

9-8792158 
4782 

7407 
8800031 
2654 
5277 
7900 

9-8810522 
3144 
5765 
8386 

9-8821007 



3627 
6246 
8866 

9-8831484 
4103 
6721 
9338 

9-8841956 
4572 
7189 
9805 

9-8852420 
5035 
7650 

9-8860264 
2878 
5492 
8105 

9-8870718 
3330 
5942 
8554 

9-8881165 
3775 
6386 
8996 

9-8891605 
4214 
6823 
9432 

9-8902040 
4647 
7254 
9861 

9-8912468 
5074 
7679 

9-8920285 
2890 
5494 
80 
52° 



dij} 

262^ 
2628 
2627 
2627 
2627 
2626 
2626 
2625 
2624 
2625 
2624 
2623 
2623 
2623 
2622 
2622 
2621 
2621 
2621 
2620 

2619 
2620 
2618 
2619 
2618 
2617 
2618 
2616 
2617 
2616 
2615 
2615 
2615 
2614 
2614 
2614 
2613 
2613 
2612 
2612 

2612 
2611 
2610 
2611 
2610 
2609 
2609 
2609 
2609 



2607 
2607 
2607 
2607 
2606 
2605 
2606 
2605 
2604 
2604 



38° 

9-8928098 

9-8930702 
3306 
5909 
8511 

9-8941114 
3715 
6317 
8918 

9-8951519 

4119 
6719 
9319 

9-8961918 
4517 
7116 
9714 

9-8972312 
4910 
7507 

9-8980104 
2700 
5296 
7892 

9-8990487 
3082 
5677 
8271 

9-9000865 
3459 

6052 
8645 

9-9011237 
3830 
6422 
9013 

9-9021604 
4195 
6786 
9376 

9-9031966 
4555 
7144 
9733 

9-9042321 
4910 
7497 

9-9050085 
2672 
5259 
7845 

9-9060431 
3017 
5603 
8188 

9-9070773 
3357 
5941 
8525 

9-9081109 
3692 
51° 



10 



2604 
26C4 
2603 
2602 
2603 
2601 
2602 
2601 
2601 
2600 
2600 
2600 
2599 
2599 
2599 
2598 
2598 
2598 
2597 
2597 
2596 
2596 
2596 
2595 
2595 
2595 
2594 
2594 
2594 
2593 

2593 
2592 
2593 
2592 
2591 
2591 
2591 
2591 
2590 
2590 

2589 
2589 
2589 

2588 
2589 
2587 
2588 
2587 
2587 
2586 

2586 

2586 
2586 
2585 
2585 
2584 
2584 
2584 
2584 
2583 



39 

9-9083692 

6275 

8858 
9-9091440 

4022 

6603 

9185 
9-9101766 

4347 

6927 

9507 
9-9112087 

4666 

7245 

9824 
9-9122403 

4981 

7559 
9-9130137 

2714 

5291 

7868 
9-9140444 

3020 

5596 

8171 
9-9150747 

3322 

5896 

8471 

9-9161045 

3618 

6192 

8765 
9-9171338 

3911 

6483 

9055 
9-9181627 

4198 

6769 

9340 
9-9191911 

4481 

7051 

9621 
9-9202191 

4760 

7329 

9898 

9-9212466 

5034 

7602 
9-9220170 

2737 

5304 

7871 
9-9230437 

3004 

5570 

8135 
50° 

LOG. COTAN. 



dlj}\ 

25S3 
2583 
2582 
2582 
2581 
25S2 
•2581 
2581 
2580 
2580 
2580 
2579 
2579 
2579 
2579 
2578 
2578 
2578 
2577 
2577 
2577 
2576 
2576 
2576 
2575 
2576 
2575 
2574 
2575 
2574 

2673 
2574 
2573 
2573 
2573 
2572 
2572 
2572 
2571 
2571 
2571 
2571 
2570 
2570 
2570 
2570 
2569 
2569 
2569 
2568 

2568 
2568 
2568 
2567 
2567 
2567 
2566 
2567 
2566 
2565 

diff. 



40° 

9-9238135 

9-9240701 
3266 
5831 
8396 

9-9250960 
3524 
6088 
8652 

9-9261215 

3778 
6341 
8904 

9-9271466 
4028 
6590 
9152 

9-9281713 
4274 
6835 

9396 
9-9291956 
4516 
7076 
9636 
9-9302195 
4755 
7314 
9872 
9-9312431 

4989 
7547 

9-9320105 
2662 
5220 
7777 

9-9330334 
2890 
5446 
8003 

9-9340559 
3114 
5670 
8225 

9-9350780 
3335 
5889 
8444 

9-9360998 
3552 

6105 
8659 

9-9371212 
3765 
6318 
8871 

9-9381423 
3975 
6527 
9079 

9-9391631 
49° 



2566 
2565 
2505 
2565 
2564 
2564 
2564 
2564 
2563 
2563 
2563 
2563 
2562 
2562 
2562 
2562 
2561 
2561 
2561 
2561 

2560 
2560 
5560 
2560 
2559 
2560 
2559 
2558 
2559 
2558 

2558 
2558 
2557 
2558 
2557 
2557 
2556 
2556 
2557 
2556 

2555 
2556 
2555 
2555 
2555 
2554 
2555 
2554 
2554 
2553 
2554 
2553 
2553 
2553 
2553 
2552 
2552 
2552 
2552 
2552 

diff. 



41° 

9-9391631 
4182 
6733 
9284 

9-9401835 
4385 
6936 
9486 

9-9412036 
4585 

7135 
9684 

9-9422233 
4782 
7331 
9879 

9-9432428 
4976 
7524 

9-9440072 

2619 
5166 
7714 

9-9450261 
2807 
5354 
7900 

9-9460447 
2993 
5539 
8084 

9-9470630 
3175 
5720 
8265 

9-9480810 
3355 
5899 
8443 

9-9490987 

3531 
6075 
8619 

9-9501162 
3705 
6248 
8791 

9-9511334 
3876 
6419 
8961 

9-9521503 
4045 
6587 
9128 

9-9531670 
4211 
6752 
9293 

9-9541834 

4374 

48° 



2551^0 
2551 ^^ 



2551 
2551 
2550 
2551 
2550 
2550 
2549 
2550 

2549 
2549 
2549 
2549 
2548 
2549 
2548 
2548 
2546 
2547 

2547 
2548 
2547 
2546 
2547 
2546 
2547 
2546 
2546 
2545 

2546 
2545 
2545 
2545 
2545 
2545 
2544 
2544 
2544 
2544 

2544 
2544 
2543 
2543 
2543 
2543 
2543 
2542 
2543 
2542 

2542 
2542 
2542 
2541 
•2542 
2541 
2541 
2541 
2541 
2540 



no 



LOG. SINE. 



[Table it. 



42° 

9-8255109 
6512 
7913 
9314 

9-8260715 
2114 
3512 
4910 
6307 
7703 

9098 
9-8270493 
1887 
3279 
4671 
6063 
7453 
8843 
9-8280231 
1619 
3006 
4393 
5778 
7163 
8547 
9930 
9-8291312 
2694 
4075 
5454 

6833 
8212 
9589 

9-8300966 
2342 
3717 
5091 
6464 
7837 
9209 

9-8310580 
1950 



3320 
4688 
6056 
7423 
8789 
9-8320155 
1519 
2883 

4246 
5609 
6970 
8331 
9691 
9-8331050 
2408 
3766 
5122 
6478 
7833 
470 



diff. 

1403 
1401 
1401 
1401 
1399 
1398 
1398 
1397 
1396 
1395 

1395 
1394 
1392 
1392 
1392 
1390 
1390 
1388 
1388 
1387 
1387 
1385 
1385 
1384 
1383 
1382 
1382 
1381 
1379 
1379 

1679 
1377 
1377 
1376 
1375 
1374 
1373 
1373 
1372 
1371 
1370 
1370 
1368 
1368 
1367 
1366 
1366 
1364 
1364 
1363 

1363 
1361 
1361 
1360 
1359 
1358 
1358 
1356 
1356 
1355 



430 

9-8337833 
9188 

9-8340541 
1894 
3246 
4597 
5948 
7297 
8646 
9994 

9-8351341 

2688 
4033 
5378 
6722 
8066 
9408 

9-8360750 
2091 
3431 
4771 
6109 
7447 
8784 

9-8370121 
1456 
2791 
4125 
5458 
6790 
8122 
9453 

9-8380783 
2112 
3441 
4769] 
6096 
7422 
8747 

9-8390072 

1396 
2719 
4041 
5363 
6684 
8004 
9323 
9-8400642 
1959 
3276 

4593 
5908 
7223 
8537 
9850 
9-8411162 
2474 
3785 
5095 
6404 
7713 
46° 



diff. 

1355 
1353 
1353 
1352 
1351 
1351 
1349 
1349 
1348 
1347 
1347 
1345 
1345 
1344 
1344 
1342 
1342 
1341 
1340 
1340 

1338 
1338 
1337 
1337 
1335 
1335 
1334 
1333 
1332 
1332 
1331 
1330 
1329 
1329 
1328 
1327 
1326 
1325 
1325 
1324 

1323 
1322 
1322 
1321 
13-20 
1319 
1319 
1317 
1317 
1317 

1315 
1315 
1314 
1313 
1312 
1312 
1311 
1310 
1309 
1309 

diff. 



440 
9-8417713 

9021 
9-8420328 

1634 

2939 

4244 

5548 

6851 

8154 

9456 

9-8430757 

2057 

3356 

4655 

5953 

7250 

8547 

9842 
9-8441137 

2432 

3725 
5018 
6310 
7601 
8891 
9-8450181 
1470 
2758 
4045 
5332 

6618 

7903 

9188 
9-8460471 

1754 

3036 

4318 

5599 

6879 

8158 

9436 
9-8470714 

1991 

3267 

4543 

5817 

7091 

8365 

963 
8480909 

2180 

3450 

4720 

5989 

7257 

8524 

9791 
9-8491057 

2322 

3586 

4850 
450 1 

LOG. COSINE. 



diff 

1308 
1307 
1306 
1305 
1305 
1304 
1303 
1303 
1302 
1301 

1300 
1299 
1299 
1298 
1297 
1297 
1295 
1295 
1295 
1293 

1293 
1292 
1291 
1290 
1290 
1289 
1288 
1287 
1287 
1286 

1285 
1285 
1283 
1283 
1282 
1282 
1281 
1280 
1279 
1278 
1278 
1277 
1276 
1276 
1274 
1274 
1274 
1272 
1272 
1271 
1270 
1270 
1269 
1268 
267 
1267 
1266 
1265 
1264 
1264 



450 

9-8494850 
6113 
7375 
8637 
9897 

9-8501157 
2417 
3675 
4933 
6190 
7446 
8702 
9957 

9-8511211 
2465 
3717 
4969 
6220 
7471 
8721 

9970 
9-8521218 
2466 
3713 
4959 
6204 
7449 
8693 
9936 
9-8531179 
2421 
3662 
4902 
6142 
7381 
8619 
9856 
9-8541093 
2329 
3564 

4799 
6033 
7266 
8499 
9730 

9-8550961 
2192 
3421 
4650 
5878 
7106 
8332 
9558 

9-8560784 
2008 
3232 
4455 
5678 
6900 
8121 
9341 
440 



diff 

1263 
1262 
1262 
1260 
1260 
1260 
1258 
1258 
1257 
1256 

1256 
1255 
1254 
1254 
1252 
1252 
1251 
1251 
1250 
1249 
1248 
1248 
1247 
1246 
1245 
1245 
1244 
1243 
1243 
1242 

1241 
1240 
1240 
1239 
1238 
1237 
1237 
1236 
1235 
1235 

1234 
1233 
1233 
1231 
1231 
1231 
1229 
1229 
1228 
1228 
1226 
1226 
1226 
1224 
1224 
1223 
1223 
1222 
1221 
1220 

diff.. 



46° 

9-8569341 

9-8570561 

1779 

2998 

4215 

5432 

6648 

7863 

9078 

9-8580292 

1505 
2718 
3929 
5141 
6351 
7561 
8770 
9978 
9-8591186 
2393 

3599 

4804 
6009 
7213 
8416 
9619 

9-8600821 
2022 
3223 
4423 
5622 
6821 
8018 
9215 

9-8610412 
1608 
2803 
3997 
5190 
6383 
7576 
8767 
9958 

9-8621148 
2338 
3526 
4714 
5902 
7088 
8274 

9460 
9-8630644 
1828 
3011 
4194 
5376 
6557 
7737 
8917 
9-8640096 
1275 



diff'. 

1220 
1218 
1219 
1217 
1217 
1216 
1215 
1215 
1214 
1213 

1213 
1211 
1212 
1210 
1210 
1209 
1208 
1208 
1207 
1206 
1205 
1205 
1204 
1203 
1203 
1202 
1201 
1201 
1200 
1199 

1199 

1197 

1197 

1197 

1196 

1195 

1194 

1193 

1193 

1193 

1191 

1191 

1190 

1190 

1188 

1188 

1: 

1186 

1186 

1186 

1184 

1184 

1183 

1183 

1182 

1181 

1180 

1180 

1179 

1179 

diff.. 



Table ii.] 



LOG. TAN. 



Ill 



9-9544374 
6915 
9455 

9-9551995 
4535 
7075 
9615 

9-9562154 
4694 
7-233 
9772 

9-9572311 
4850 
7389 
9927 

9-9582465 
5004 
7542 

9-9590080 
2618 

5155 
7693 

9-9600230 
2767 
5305 
7842 

9-9610378 
2915 
5452 
7988 

9-9620525 
3061 
5597 
8133 

9-9630669 
3204 
5740 
8275 

9-9640811 
3346 
5881 
8416 

9-9650951 
3486 
6020 
8555 

9-9661089 
3623 
6157 
8692 

9-9671225 
3759 
6293 
8827 

9-9681360 
3893 
6427 
8960 

9-9691493 

4026 

6559 

470 



diff. 

2541 
2540 
2540 
2540 
2540 
2540 
2539 
2540 
2539 
2539 
2539 
2539 
2539 
2538 
2538 
2539 
2538 
2538 
2538 
2537 

2538 
537 
2537 
253C 
2537 
2536 
2537 
2537 
2536 
2537 

2536 
2536 
2536 
2536 
2535 
2536 
2535 
2536 
2535 
2535 

2535 
2535 
2535 
2534 
2535 
2534 
•2534 
2534 
2535 
2533 
2534 
2534 
2534 
2533 
2533 
2534 
2533 
2533 
2533 
2533 

diff. 



430 

9-9696559 
9091 

9-9701624 
4157 
6689 
9221 

9-9711754 
4286 
6818 
9350 

9-9721882 
4413 
6945 
9477 

9-9732008 
4539 
7071 
9602 

9-9742133 
4664 

7195 
9726 

9-9752257 
4787 
7318 
9849 

9-9762379 
4909 
7440 
9970 

9-9772500 
5030 
7560 

9-9780090 
2620 
5149 
7679 

9-9790209 
2738 
5268 
7797 

9-9800326 
2856 
5385 
7914 

9-9810443 
2972 
5501 
8030 

9-9820559 

3087 
5616 
8145 

9-9830673 
3202 
5730 
8259 

9-9840787 

3315 

5844 

8372 

46° 



diff. 

2532 
2533 
2533 
2532 
2532 
2533 
2532 
2532 
2532 
2532 

2531 
2532 
2532 
2531 
2531 
2532 
2531 
2531 
2531 
2531 
2531 
2531 
3530 
2531 
'2531 
2530 
2530 
2531 
2530 
2530 
2530 
2530 
2530 
2530 
2529 
2530 
2530 
2529 
2530 
2529 

2529 
2530 
2529 
2529 
2529 
2529 
2529 
2529 
2529 
2528 

2529 
2529 
2528 
2529 
2528 
2529 
2528 
2528 
2529 
2528 



44° 
9-9848372 
9-9850900 

3428 

5956 

8484 
9-9861012 

3540 

6068 

8596 
9-9871123 

3651 

6179 

8706 
9-9881234 

3761 

6289 

8816 
9-9891344 

3871 

6399 

8926 
9-9901453 

3981 

6508 

9035 
9-9911562 

4089 

6616 

9143 
9-9921670 

4197 

6724 

9251 
9-9931778 

4305 

6832 

9359 
9-9941886 

4413 

6940 

9466 
9-9951993 

4520 

7047 

957 
9-9962100 

4627 

7154 

9680 
9-9972207 

4734 

7260 

9787 
9-9982314 

4840 

7367 

9893 
9-9992420 

4947 

7473 
100000000 
450 

LOG. COTAN. 



2528 
2528 

2528 
2528 
2528 
2528 
2528 
2528 
2527 
2528 

2528 
2527 
2528 
2527 
2528 
2527 
2528 
2527 
2528 
2527 

2527 
2528 
2527 
2527 
2527 
2527 
2527 
2527 
2527 
2527 

2527 
2527 
2527 
2527 
2527 
2527 
2527 
2527 
2527 
2526 
2527 
2527 
2527 
2526 
25-27 
2527 
2527 
2526 
2527 
2527 

2526 
2527 
2527 
2526 
2527 
2526 
2527 
2527' 
2526 
2527 

diff 



450 

100000000 

2527 
5053 
7580 

10'0010107 
2633 
5160 
7686 

10-0020213 
2740 
5266 
7793 

10-0030320 
2846 
5373 
7900 

100040427 
2953 
5480 
8007 

10-0050534 
3060 
5587 
8114 

10-0060641 
3168 
5695 
8222 

10-0070749 
3276 
5803 
8330 

100080857 
3384 
5911 
8438 

10-0090965 
3492 
6019 
8547 

100101074 
3601 
6129 
8656 

100111184 
3711 
6239 
8766 

10-0121294 
3821 

6349 
8877 

10-0131404 
3932 
6460 
8988 

100141516 
4044 
6572 
9100 

10-01516-28 
440 



diff. 

2527 
2526 
2527 
2527 
2526 
2527 
2526 
2527 
25-27 
2526 

2527 
2527 
2526 
2527 
2527 
2527 
2526 
2527 
2527 
2527 

2526 
25-27 
2527 
2527 
2527 
2527 
2527 
2527 
2527 
2527 
2527 
2527 
2527 
2527 
2527 
•2527 
2527 
2527 
2528 
2527 

2527 
2528 
2527 
2528 
2527 
2528 
2527 
2528 
2527 
2528 

2528 
2527 
■2528 
2528 
2528 
2528 
2528 
2528 
2528 
2528 

diff 



46° 1 
10-01516-28 
4156 
6685 
9213 
1001'61741 
4270 
6798 
9327 
10-0171855 
4384 

6913 
9441 

10-0181970 
4499 
7028 
9557 

10-0192086 
4615 
7144 
9674 

10-0202203 
4732 
7262 
9791 

10-021-2321 
4851 
7380 
9910 

100222440 
4970 
7500 

10-0230030 
2560 
5091 
7621 

10-0240151 
2682 
5213 
7743 

10-0250274 

2805 
5336 
7867 

10-0260398 
2929 
5461 
7992 

10-0270523 
3055 
5587 

8118 

10-0280650 

3182 

5714 

8246 

10-0290779 

3311 

5843 

8376 

IO-O3C0909 

3441 

43° 



diff 

2528 
25-29 
2528 
2528 
2529 
2528 
2529 
2528 
2529 
2529 
2528 
2529 
2529 
2529 
2529 
2529 
2529 
2529 
2530 
2529 

2529 
2530 
2529 
2530 
2530 
2529 
2530 
2530 
2530 
2530 
2530 
2530 
2531 
2530 
2530 
2531 
2531 
2530 
2531 
2531 

2531 
2531 
2531 
2531 
2532 
2531 
2531 
2532 
2532 
2531 
2532 
2532 
2532 
2532 
2533 
2532 
2532 
2533 
2533 
2532 

diff 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 





112 



LOG. SINE. 



[Table ii. 



470 

9-8641275 
2452 
3629 
4806 
5981 
7156 
8331 
9504 
•8650677 
1849 
3021 
4192 
5362 
6531 
7700 
8868 

9-8660036 
1203 
2369 
3534 

4699 
5863 
7026 
8189 
9351 
9-3670512 
1673 
2833 
3992 
5151 

6309 
7466 
8623 
9779 
9-8680934 
2088 
3242 
4396 
5548 
6700 

7851 
9002 
9-8690152 
1301 
2449 
3597 
4744 
5891 
7037 
8182 

93-26 

9-8700470 

1613 

2756 

3898 

5039 

6179 

7319 

8458 

9597 

9-8710735 

42° 



diff. 

1177 
1177 
1177 
1175 
1175 
1175 
1173 
1173 
1172 
1172 
1171 
1170 
1169 
1169 
1168 
1168 
1167 
1166 
1165 
1165 
1164 
1163 
1163 
1162 
1161 
1161 
1160 
1159 
1159 
1158 

1157 
1157 
1156 
1155 
1154 
1154 
1154 
1152 
1152 
1151 
1151 
1150 
1149 
1148 
1148 
1147 
1147 
1146 
1145 
1144 

1144 
1143 
1143 
1142 
1141 
1140 
1140 
1139 
1139 
1138 

diff. 



48° 

-8710735 
1872 
3008 
4144 
5279 
6414 
7548 
8681 
9813 

•8720945 

2076 
3207 
4337 
5466 
6594 
7722 
8849 
9976 
-8731102 
2227 

3352 
4476 
5599 
6722 
7844 
S965 

-8740085 
1205 
2325 
3443 
4561 
5679 
6795 
7912 
9027 

-8750142 
1256 
2369 
3482 
4594 

5706 
6816 
7927 
9036 

-8760145 
1253 
2361 
3468 
4574 
5680 
6785 
7889 
8993 

-8770096 
1198 
2300 
3401 
4501 
5601 
6700 
7799 
41° 



diff. 

1137 
1136 
1136 
1135 
1135 
1134 
1133 
1132 
1132 
1131 
1131 
1130 
1129 
11-28 
1128 
1127 
1127 
1126 
1125 
1125 
1124 
1123 
1123 
1122 
1121 
1120 
1120 
1120 
1118 
1118 

1118 
1116 
1117 
1115 
1115 
1114 
1113 
1113 
1112 
1112 
1110 

nil 

1109 
1109 
1108 
1108 
1107 
1106 
1106 
1105 
1104 
1104 
1103 
1102 
1102 
1101 
1100 
1100 
1099 
1099 

diff 



490 

9-8777799 
8896 
9994 

9-8781090 
2186 
3281 
4376 
5470 
6563 
7656 

8748 
9840 
790930 
2021 
3110 
4199 
5287 
6375 
7462 
8548 
9634 
9-8800719 
1603 
2887 
3970 
5052 
6134 
7215 
8296 
9376 

9-8810455 
1534 
2612 
3689 
4766 
5842 
6918 
7992 
9067 

9-8820140 
1213 
2285 
3357 
4428 
5499 
6568 
7638 
87061 
9774 

9-8830841 

1908 
2974I 
4039 
5104 
6168 
7232 
8294 
9357 
9-8840418 
1479 
2540 
40° 



diff. 

1097 
1098 
1096 
1096 
1095 
1095 
1094 
1093 
1093 
1092 
1092 
1090 
1091 
1089 
1089 
1088 
1088 
1087 
1086 
1086 

1085 
1084 
1084 
1083 
1082 
1082 
1081 
1081 
1080 
1079 

1079 
1078 
1077 
1077 
1076 
1076 
1074 
1075 
1073 
1073 
1072 
1072 
1071 
1071 
1069 
1070 
1068 
1068 
1067 
1067 
1066 
1065 
1065 
1064 
1064 
1062 
1063 
1061 
1061 
1061 

diff. 



50° 

-8342540 
3599 
4659 
5717 
6775 
7832 
8889 
9945 

•8851000 
2055 

3109 
4162 
5215 
6267 
7319 
8370 
9420 

•8860470 
1519 
2568 
3616 
4663 
5710 
6756 
7801 
8846 
9890 

-8870934 
1977 
3019 
4061 
5102 
6142 
7182 
8221 
9260 

-8880298 
1335 
2372 
3408 
4444 
5479 
6513 
7547 
8580 
9612 

1-8890644 
1675 
2706 
3736 

4765 
5794 
6822 
7850 
8877 
9903 
1-8900929 
1954 
2979 
4003 
5026 
39° 



1059 
1060 
1058 
1058 
1057 
1057 
1056 
1055 
1055 
1054 

1053 
1053 
1052 
1052 
1051 
1050 
1050 
1049 
1049 
1048 
1047 
1047 
1046 
1045 
1045 
1044 
1044 
1043 
1042 
1042 
1041 
1040 
1040 
1039 
1039 
1038 
1037 
1037 
1036 
1036 

1035 
1034 
1034 
1033 
1032 
1032 
1031 
1031 
1030 
1029 
1029 
1028 
1028 
1027 
1026 
1026 
1025 
1025 
1024 
1023 

diff. 



51° 

9-8905026 
6049 
7071 
8092 
9113 

9-8910133 
1153 
2172 
3191 
4-208 

5226 
6242 
7258 
8274 
9289 

9-8920303 
1316 
2329 
3342 
4354 
5365 
6375 
7385 
8395 
9404 

9-8930412 
1419 
2426 
3433 
4439 
5444 
6448 
7452 
8456 
9458 

9-8940461 
1462 
2463 
3464 
4463 

5463 
6461 
7459 
8457 
9453 

9-8950450 
1445 
2440 
3435 
4429 
5422 
6414 
7406 
8398 
9389 

9-8960379 
1369 
2358 
3346 
4334 
5321 



diff. 

1023 
1022 
1021 
1021 
1020 
1020 
1019 
1019 
1017 
1018 

1016 
1016 
1016 
1015 
1014 
1013 
1013 
1013 
1012 
1011 

1010 
1010 
1010 
1009 
1008 
1007 
1007 
1007 
1006 
1005 
1004 
1004 
1004 
1002 
1003 
1001 
1001 
1001 
999 
1000 



996 
997 
995 
995 
995 
994 
993 

992 
992 
992 
991 
990 
990 
989 



987 
diff. 



LOG. COSINE. 



Table ii.] 



LOG. TAN. 



113| 



10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
!24 
|25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



470 

10-0303441 
5974 
8507 

100311040 
3573 
6107 
8640 

10-0321173 
3707 
6241 
8775 

10-0331308 
3843 
6377 
8911 

100341445 
3980 
6514 
9049 

10-0351584 

4119 
6654 
9189 

100361725 
4260 
6796 
9331 

10-0371867 
4403 
6939 
9475 

100382012 
4548 
7085 
9622 

10-0392158 
4695 
7233 
9770 

10-0402307 

4845 
7382 
9920 

100412458 
4996 
7535 

10 0420073 
2611 
5150 
7689 

10-0430228 
2767 
5306 
7846 

100440385 
2925 
5465 
8005 

100450545 
3085 
5626 



2533 
2533 
2533 
2533 
2534 
2533 
25^3 
2534 
2534 
2534 
2533 
2535 
2534 
2534 
2534 
2535 
2534 
2535 
2535 
2535 

2535 
2535 
2536 
2535 
2536 
2535 
2536 
2536 
2536 
2536 
2537 
2536 
2537 
2537 
2536 
2537 
2538 
2537 
2537 
2538 
2537 
2538 
2538 
2538 
2539 
2538 
2538 
2539 
2539 
2539 
2539 
2539 
2540 
2539 
2540 
2540 
2540 
2540 
2540 
2541 

diff. 



48° 

10-0455626 
8166 

100460707 
3248 
5789 
8330 

10-0470872 
3413 
5955 
8497 

10-0481039 
3581 
6124 
8666 

10-0491209 
3752 
6295 
8838 

10-0501381 
3925 

6469 
9013 

10-0511557 
4101 
6645 
9190 

100521735 
4280 
6825 
9370 

10-0531916 
4461 
7007 
9553 

10-0542100 
4646 
7193 
9739 

100552286 
4834 
7381 
9928 

100562476 
5024 
7572 

100570121 
2669 
5218 
7767 

100580316 
2865 
5415 
7964 

100590514 
3064 
5615 
8165 

100600716 

3267 

5818 

8369 

41° 



2540 
2541 
2541 
2541 
2541 
2542 
2541 
2542 
2542 
2542 

2542 
2543 
2542 
2543 
2543 
2543 
2543 
2543 
2544 
2544 

2544 
2544 
2544 
2544 
2545 
2545 
2545 
2545 
2545 
2546 
2545 
2546 
2546 
2547 
2546 
2547 
2546 
2547 
2548 
2547 
2547 
2548 
2548 
2548 
2549 
2548 
2549 
2549 
2549 
2549 
2550 
2549 
2550 
2550 
2551 
2550 
2551 
2551 
2551 
2551 

diff. 



490 

100608369 

100610921 

3473 

6025 

8577 
100621129 

3682 

6235 

8788 
00631341 

3895 

6448 

9002 
100641556 

4111 

6665 

9220 
10-0651775 

4330 

6886 

9441 
10-0661997 

4554 

7110 

9666 
100672223 

4780 

7338 

9895 
10-0682453 

5011 

7569 
100690128 

2686 

5245 

7805 
100700364 

2924 

5484 

8044 
10-0710604 

3165 

5726 

8287 
100720848 

3410 

5972 

8534 
100731096 

3659 

6222 

8785 
100741348 

3912 

6476 

9040 
10-0751604 

4169 

6734 

9299 
100761865 
40° 

LOG. COTAN. 



2552 
2552 
2552 
2552 
2552 
2553 
2553 
2553 
2553 
2554 

2553 
2554 
2554 
2555 
2554 
2555 
2555 
2555 
2556 
2555 

2556 
2557 
2556 
2556 
2557 
2557 
2558 
2557 
2558 
2558 
2558 
2559 
2558 
2559 
2560 
2559 
2560 
2560 
2560 
2560 
2561 
2561 
2561 
2561 
2562 
2562 
2562 
2562 
2563 
2563 
2563 
2563 
2564 
2564 
2564 
2564 
2565 
2565 
2565 
2566 

diff. 



50° 

10'0761865 
4430 
6996 
9563 

100772129 
4696 
7263 
9830 

10-0782398 
4966 

7534 
100790102 
2671 
5240 
7809 
100800379 
2949 
5519 



10-0810660 

3231 
5802 
8373 

10-0820945 
3517 
6089 
8662 

100831235 
3808 
6382 
8955 

10-0841529 
4104 
6678 
9253 

100851829 
4404 
6980 
9556 

10-0862132 

4709 
7286 
9863 

10-0872441 
5019 
7597 

100880176 
2755 
5334 
7913 

100890493 
3073 
5653 
8234 

10 0900815 
3397 
5978 
8560 

100911142 
3725 
6308 



diff 

2565 
2566 
2567 
2566 
2567 
2567 
2567 
2568 
2568 
2568 
2568 
2569 
2569 
2569 
2570 
2570 
2570 
2570 
2571 
2571 



2571 
2571 

2572 
2572 
2572 
2573 
2573 
2573 
2574 
2573 
2574 
2575 
2574 
2575 
2576 
2575 
2576 
2576 
2576 
2577 
2577 
2577 
2578 
2578 
2578 
2579 
2579 
2579 
2579 
2580 
2580 
2580 
2581 
2581 
2582 
2581 
2582 
2582 
2583 
2583 

diff. 



51° 

100916308 
8891 

100921475 
4059 
6643 
9227 

100931812 
4397 
6983 
9569 

10-0942155 
4741 
7328 
9915 

100952503 
5090 
7679 

10-0960267 
2856 
5445 
8034 

100970624 
3214 
5805 
8396 

100980987 
3578 
6170 
8763 

100991355 

3948 
6541 
9135 

10-1001729 
4323 
6918 
9513 

101012108 
4704 
7300 
9896 

10-1022493 
5090 
7688' 

10-1030286 
2884 
5483 
8082 

10-1040681 
3281 
5881 
8481 

10-1051082 
2683 
6285 



10-1061489 
4091 
6694 
9298 

101071902 
38° 



diff.\ 

25831 
2584 
2584 
2584 
2584 
2585 
2585 
2586 
2586 
2586 

2586 
2587 
2587 
2588 
2587 
2589 
2588 
2589 
2589 
2589 
2590 
2590 
2591 
2591 
2591 
2591 
2592 
2593 
2592 
2593 

2593 
2594 
2594 
2594 
2595 
2595 
2595 
2596 
2596 
2596 
2597 
2597 
2598 
2598 
2598 
2599 
2599 
2599 
2600 
2600 
2600 
2601 
2601 
2602 
2601 
2603 
2602 
2603 
2604 
2604 



57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 




10* 



114 



LOG. SINE. 



[Table 11. 



52° 

9-896 5321 
6308 
7294 
8280 
9265 

9-897 0249 
1233 
2216 
3199 
4181 
5162 
6143 
7123 
8103 
9082 

9-898 0060 
1038 
2015 
2992 
3968 
4944 
5919 
6893 
7867 
8840 
9812 

9-899 0784 
1756 
2727 
3697 
4667 
5636 
6604 
7572 
8539 
9506 

9-900 0472 
1438 
2403 
3367 
4331 
5294 
6257 
7219 
8181 
9142 

9-901 0102 
1062 
2021 
2980 
3938 
4895 
5852 
6808 
7764 
8719 
9674 

9-902 0628 
1581 
2534 
3486 
37° 



979 
978 
978 
977 
977 
976 
976 
975 
974 
974 
973 
972 
972 
972 
971 
970 
970 
969 
968 
968 
967 
967 
966 
966 
965 
964 
964 
963 
963 
962 
962 
961 
960 
9601 
959 
959 
958 
957 
957 
956 
956 
955 
955 
954 
953 
953 
952 

dif. 



53° 

1-902 3486 
4438 
5369 
6339 
7289 
6239 
9188 

1-903 0136 
1084 
2031 
2977 
3923 
4868 
5813 
6757 
7701 
8644 
9587 

'•904 0529 
1470 
2411 
3351 
4291 
5230 
6168 
7106 
8043 
8980 
9916 

-905 0852 
1787 
2722 
3656 
4589 
5522 
6454 
7386 
8317 
9247 

•906 0177 
1107 
2036 
2964 
3892 
4819 
5745 
6671 
7597 
8522 
9446' 

•907 0370| 
1293 
2216 
3138| 
4059i 
4930 
59011 
6820' 
7740' 
8658 
9576 
36° 



dif. 

952 
951 
950 
950 
950 
949 
948 
948 
947 
946 
946 
945 
945 
944 
944 
943 
943 
942 
941 
941 
940 
940 
939 
938 
938 
937 
937 
936 
936 
935 
935 
934 
933 
933 
932 
932 
931 
930 
930 
930 

929 
928 
928 
927 
926 
926 
9-26 
925 
9-24 
9'24 

923 
923 
922 
921 
921 
921 
919 
920 
918 
918 

dif. 



54° dif. 
9-907 9576 
9-9C8 C494 

1411 

2327 

3243 

4159 

5073 

5988 

6901 

7814 

8727 

9639 
9-909 0550 

1461 

2371 

3281 

4190 

5099 

600 

6915 

7821 

8728 

9634 
9-910 0539 

1444 

2348 

3251 

4155 

5057 

5959 

6860 

7761 

8661 

9561 
9-911 0460 

1359 

2257 

3155 

4051 

4948 

5844 

6739 

7634 

8528 

9422 
9-912 0315 

1207 

2099 

2991 

3882 

4772 

5662 

6551 

7440 

8328 

9215 
9-913 0102 

0989 

1875 

2760 

3645 
35° 



55° 

9-913 3645 
4530 
5413 
6296 
7179 
8061 
8943 
9824 

9-914 0704 
1584 

2464 
3342 
4221 
5099 
5976 
6852 
7729 
8604 
9479 

9-915 0354 
1228 
2101 
2974 
3846 
4718 
5589 
6460 
7330 
8200 
9069 
9937 

19-916 0805 
1673 
2539 
3406 
4272 
5137 
6002 
6866 
7730 
6593 
9455 

9-917 0317 
1179 
2040 
2900 
3760 
4619 
5478 
6336 
7194 
8051 
8908 
9764 

9-918 0620 
1475 
2329 
3183 
4037 
4890 
5742 



dif 

LOG. COSINE. 



34° 



868 
866 
867 
366 
865 
865 
364 
864 
863 
862 
862 
862 
861 
360 
860 
859 
859 
858 
858 
857 
857 
856 
856 
855 
854 
854 
854 
853 
852 

dif 



56° 
918 5742 
6594 
7445 
8296 
9146 
9996 

9-919 0845 
1694 
2542 
3390 
4237 
5083 
5929 
677 
7619 
8464 
9308 
920 0151 
0994 
1836 
2678 
3519 
4360 
5200 
6039 
6878 
7717 
8555 
9393 

9-921 0-229 

1066 
1902 
2737 
3572 
4406 
5-240 
6073 
6906 
7738 
8570 
9401 

9^922 0232 
1062 
1891 
2721 
3549 
4377 
5205 
6032 
6858 
7684 
8509 
9334 

9-923 0158 
0982 
1805 
2628 
3450 
4272 
5093 
5914 
33° 



dif 

852 
851 
851 
850 

849 
849 
848 
848 
847 
846 
846 
846 
844 
8-45 
844 
343 
843 
342 
842 
841 
341 
340 
339 
339 
839 



836 
837 
836 
835 
835 
834 
834 
833 
833 
832 
832 
831 
831 
830 
29 
830 
328 
828 
828 
827 
826 
826 
825 
825 
824 
824 
823 
823 
822 
822 
821 
821 

dif. 



Table ii.] 



LOG. TAN. 



115 



10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



52^ 

10-1071902 
4506 
7110 
9715 

10-1082321 
4926 
7532 

101090139 
2746 
5353 
7960 

10-1100568 
3177 
5786 
8395 

10-1111004 
3614 
6225 
8835 

10-1121446 
4058 
6670 
9282 

10-1131895 
4508 
7122 
9736 

10-1142350 
4965 
7580 

10-1150195 
2811 
5428 
8044 

10-1160662 
3279 
5897 
8516 

10-1171134 
3754 

6373 
8993 

10-1181614 
4235 
6856 
9478 

10-1192100 
4723 
7346 
9969 

10-1202593 
5218 
7842 

101210467 
3093 
5719 
8346 

10-1220973 
3600 
6228 



370 



diff. 

2604 
2604 
2605 
2606 
2605 
3606 
2607 
2607 
2607 
2607 
2608 
2609 
2609 
2609 
2609 
2610 
2611 
2610 
2611 
2612 

2612 
2612 
2613 
2613 
2614 
2614 
2614 
2615 
2615 
2615 
2616 
2617 
2616 
2618 
2617 
2618 
2619 
2618 
2620 
2619 
2620 
2621 
2621 
2621 
2622 
2622 
2623 
2623 
2623 
2624 

2625 
2624 
2625 
2626 
2626 
2627 
2627 
2627 
2628 
2628 



530 

101228856 

J01231485 
4114 
6743 
9373 

10-1242004 
4635 
7266 
9898 

10-1252530 
5162 
7796 

10-1260429 
3063 
5698 
8332 

10-1270968 
3604 
6240 
8877 

10-12S1514 
4152 
6790 
9428 

10-1292067 
4707 
7347 
9987 

10-1302628 
5269 
7911 

10-1310554 
3196 
5840 
8483 

10-1321127 
3772 
6417 
9063 

10-1331709 
4356 
7003 
9650 

10-1342298 
4947 
7596 

10-1350245 
2895 
5546 
8197 

10-136 

3500 
6152 
8805 

10-1371459 
4113 
6767 
9422 

10-1382077 
4733 
7390 
36° 



dijf- 

2629 
2629 
2629 
2630 
2631 
2631 
2631 
2632 
2632 
2632 
2634 
2633 
2634 
2635 
2634 
2636 
2636 
2636 
2637 
2637 

2638 
2938 
2638 
2639 
?640 
2640 
2640 
2641 
2641 
2642 

2643 
2642 
2644 
2643 
2644 
2645 
2645 
2646 
2646 
2647 
2647 
2647 
2648 
2649 
2649 
2649 
2650 
2651 
2651 
2651 
2652 
2652 
2653 
2654 
2654 
2654 
2655 
2655 
2656 
2657 

Uiif} 



54° 

101387390 

10-1390046 
2704 
5362 
8020 

10-1400679 
3339 
5998 
8659 

101411320 

3981 
6643 
9306 

10-1421969 
4632 
7296 
9961 

10-1432626 
5292 
7958 

10-1440624 
3292 
5959 
8628 

10-1451296 
3966 
663 
9306 

10-1461977 
4648 
7320 
9992 

10-1472665 
5339 
8013 

10-1480688 
3363 
6039 
8715 

10-1491392 

4069 
6747 
9425 

10-1502104 
4784 
7464 

10-1510145 
2826 
5508 
8190 

10-1520873 
3556 
6240 
8925 

10-1531610 
4295 
6982 
9668 

10-1542356 
5044 
7732 
350 



diff. 

2656 
2658 
2658 
2658 
2659 
266U 
2659 
2661 
2661 
2661 
2662 
2663 
2663 
2663 
2664 
2665 
2665 
2666 
2666 
2666 

2668 
2667 
2669 
2668 
2670 
2669 
2671 
2671 
2671 
2672 

2672 
2673 
2674 
2674 
2675 
2675 
2676 
2676 
2677 
267' 
2678 
2678 
2679 
2660 
2680 
2681 
2681 
2682 
2682 
2683 



55° 

10.1547732 

10-1550421 
3111 
5801 
8492 

10-1561183 
3875 
6568 
9261 

10-1571954 



diJi: 

2689 
2690 
2690 
2691 
2691 
2692 
2693 
2693 
2693 
2695 



4649 
7343 

101580039 
2735 
5431 
8129 

101590826 
3525 
6224 
8923 

10-1601623 
4324 
7025 
9727 

101612429 
5133 
7836 

10 ■1620540 
3245 
5951 
8657 

10-1631364 
4071 
6779 
9487 

10-1642196 
4906 
7616 

10-1650327 
3039 
5751 
8464 

lC-1661177 
3891 
6606 
9321 

10-1672037 
4754 
7471 

10-1680189 
2907 
5626 
8346 

10-1691066 
3787 
6508 
9231 

10-1701953 

4677 

9fiRRl 7401 

^""" 10-1710126 

diff\ 34" 



2683 
2684 
2685 
2685 
2685 
2687 
2686 
2688 



2694 
2696 
2696 
2696 
2698 
2697 
2699 
2699 
2699 
2700 

2701 
2701 
2702 
2702 
2704 
2703 
2704 
2705 
2706 
2706 

2707 
2707 
2708 
2708 
2709 
2710 
2710 
2711 
2712 
2712 
2713 
2713 
2714 
2715 
2715 
2716 
2717 
2717 
2718 
2718 

2719 
2720 
2720 
2721 
2721 
2723 
2722 
2724 
2724 
2725 

did 



56° 

10-1710126 
2851 
5577 
8304 

101721031 
3759 
6487 
9217 

101731947 
4677 

7408 

10-1740140 
2873 
5606 
8340 

10-1751074 
3809 
6545 
9281 

101762019 
4756 
7495 

10-1770234 
2974 
5714 
8455 

10-1781197 
3940 
6683 
9426 

10-1792171 
4916 
7662 

10-1800408 
3156 
5904 
8652 

10-1811401 
4151 
6902 
9653 

10-1822405 
5158 
7911 

10-1830665 
3420 
6176 
8932 

10-1841689 
4446 

7205 
9964 

10-1852723 
5484 
8245 

10-1861007 
3769 
6532 
9296 

10-1872061 
4826 
330 



diff. 

2725 
2726 
2727 
2727 
2728 
2728 
2730 
2730 
2730 
2731 
2732 
2733 
2733 
2734 
2734 
2735 
2736 
2736 
2738 
2737 

2739 
2739 
2740 
2740 
2741 
2742 
2743 
2743 
2743 
2745 

2745 
2746 
2746 
2748 
2748 
2748 
2749 
2750 
2751 
2751 
2752 
2753 
2753 
2754 
2755 
2756 
2756 
2757 
2757 
2759 
2759 
2759 
2761 
2761 
2762 
2762 
2763 
2764 
2765 
2765 



LOG. COTAN. 



116 



LOG. SINE. 



[Table 



57° 

9-923 5914 
6734 
7554 
8373 
9191 

9-924 0010 
0827 
1644 
2461 
3277 

4092 
4907 
5721 
6535 
7349 
8161 
8974 
9786 

9-925 0597 
1408 
2218 
3028 
383 
4646 
5454 
6261 
7069 
7875 
8681 
9487 

9-926 0292 
1096 
1901 
2704 
3507 
4310 
5112 
5913 
6714 
7514 

8314 
9114 
9913 
9-927 0711 
1509 



2306 
3103 
3899 
4695 
5490 
6285 
7079 
7873 
8666 
9459 
9-928 0251 
1043 
1834 
2625 
3415 
4205 
320 



dif. 

820 
820 
819 
818 
819 
817 
817 
817 
816 
815 
815 
814 
814 
814 
812 
13 
812 
811 
811 
810 
810 
809 
809 
808 
807 
808 
806 
806 
806 
805 
804 
805 
803 
803 
803 
802 
80L 
801 
800 
800 
800 
799 
798 
798 
797 
797 
796 
796 
795 
795 
794 
794 
793 
793 
792 
792 
791 
791 
790 
790 



58° 

9-928 4205 
4994 
5783 
6571 
7358 
8145 
8932 
971 

9-929 05U4 
1289 
2073 
2857 
3641 
44-24 
5207 
5989 
6770 
7551 
8332 
9112 

9891 
9-930 0670 
1448 
2226 
3004 
3781 
4557 
5333 
6109 
6883 

7658 
8432 
9205 
9978 

9-931 0750 
1522 
2294 
3065 
3835 
4605 
5374 
6143 
6911 
7679 
8447 
9213 
9980 

9-932 0746 
1511 
2276 
3040 
3804 
4567 
5330 
6092 
6854 
7616 
8376 
9137 
9897 

9-933 0656 
31° 



dif. 

789 
789 
788 
787 
787 
787 
786 
786 
785 
784 
784 
784 
783 
783 
782 
781 
781 
781 
780 
779 
779 
778 
778 
778 
777 
776 
776 
776 
774 
775 
774 
773 
773 
772 
772 
772 
771 
770 
770 
769 
769 
768 
768 
768 
766 
767 
766 
765 
765 
764 
764 
763 
763 
762 
762 
762 
760 
761 
760 
759 

dif. 



59° 
9-933 0656 

1415 

2173 

2931 

3688 

4445 

5201 

5957 

6713 

7467 

8222 

8976 

9729 
9-934 0482 

1234 

1986 

2737 

3488 

4238 

4988 

5738 

6486 

7235 

7983 

8730 

9477 
9*9350223 

0969 

1715 

2459 

3204 

3948 

4691 

5434 

6177 

6918 

7660 

8401 

9141 

9881 
9-936 0621 

1360 

2098 

2836 

3574 

4311 

5047 

5783 

6519 

7254 

7988 
8722 
9456 
9-9370189 
0921 
1653 
2385 
3116 
3847 
4577 
5306 
30° 

LOG. COSINE. 



dij-. 

759 

758 
758 
757 
757 
756 
756 
756 
754 
755 
754 
753 
753 
752 
752 
751 
751 
750 
750 
750, 
748* 
749 
748 
747 
747 
746 
746 
746 
744 
745 
744 
7431 
743 
743 
741 
742 
741 
740 
740 
740 
739 
738 
738 
738 
737 
736 
36 
736 
735 
734 
734 
734 
733 
732 
732 
732 
731 
731 
730 
729 

diff. 



60° 

9-937 5306 
6035 
6764 
7492 
8220 
8947 
9674 

9-9380400 
1126 
1851 
2576 
3300 
4024 
4747 
5470 
6192 
6914 
7635. 
8356 
9076 
9796 

9-9390515 
1234 
1953 
2671 
3388 
4105 
4821 
5537 
6253 
6968 
7682 
8396 
9110 
9823 

9-940 0535 
1248 
1959 
2670 
3381 
4091 
4801 
5510 
6219 
6927 
7634 
8342 
9048 
9755 

9-941 0461 
1166 
1871 
2575 
3279 
3982 
4685 
5388 
6090 
6791 
7492 
8193 



729 
729 
728 
728 
727 
727 
726 
726 
725 
725 
724 
724 
723 
723 
722 
722 
721 
721 
720 
720 
719 
719 
719 
718 
717 
717 
716 
716 
716 
715 
714 
714 
714 
713 
712 
713 
711 
711 
711 
710 
710 
709 
709 
708 
707 
708 
706 
707 
706 
705 
705 
704 
704 
703 
703 
703 
702 
701 
701 
701 

dif. 



61° 

9-941 8193 
8893 
9592 

9-942 0291 
0990 
1688 
2386 
3083 
3779 
4476 
5171 
5866 
6561 
7255 
7949 
8643 
9335 

9-9430028 
0720 
1411 
2102 
2792 
3482 
4172 
4861 
5549 
6238 
6925 
7612 
8299 
8985 
9671 

9-944 0356 
1041 
1725 
2409 
3092 
3775 
4457 
5139 
5821 
6501 
7182 
7862 
8541 
9220 
9899 

9-945 0577 
1255 
1932 
2609 
3285 
3960 
4636 
5310 
5985 
6659 
7332 
8005 
8677 
9349 
28° 



dif 

700 
699 
699 
699 
698 
698 
697 
696 
697 
695 



687 
687 
687 
686 
686 
685 
685 
684 
684 
683 
683 
682 
682 
682 

680 
681 
680 
679 
679 
679 
678 
678 
677 
677 
676 
675 
676 
674 
675 
674 
673 
673 
672 
672 

dif. 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 



Table ii.] 



LOG. TAN. 



117 



57° 

101874826 
7592 

101880359 
3127 
5895 
8664 

10^1891434 
4204 
6975 
9747 

101902520 
5293 
8067 

10-1910842 
3617 
6394 
9171 

10-1921948 
4727 
7506 

10-1930286 
3067 
5848 
8630 

10-1941413 
4197 
6981 
9767 

10-1952553 
5339 
8127 

10-1960915 
3704 
6494 
9284 

10-1972075 
4867 
7660 

10-1980454 
3248 
6043 
8839 

10-1991635 
4433 
7231 

10-2000030 
2830 
5630 
8431 

10-2011233 
4036 
6840 
9644 

10-2022449 
5255 
8062 

10-2030870 
3678 
6487 
9297 

10-2042108 
32° 



diff. 

2766 
2767 
2768 
2768 
2769 
2770 
2770 
2771 
2772 
2773 
2773 
2774 
2775 
2775 
2777 
2777 
2777 
2779 
2779 
2780 
2781 
2781 
2782 
2783 
2784 
2784 
2786 
2786 
2786 
2788 
2788 
2789 
2790 
2790 
2791 
2792 
2793 
2794 
2794 
2795 
2796 
2796 
2798 
2798 
2799 
2800 
2800 
2801 
2802 
2803 
2804 
2804 
2805 
2806 
2807 
2808 
2808 
2809 
2810 
2811 

diff. 



58 

iO-2042108 
4919 
7732 

10-2050545 
3359 
6173 
8989 

10-2061805 
4622 
7440 

10-2070259 
3079 
5899 
8720 

10-2081542 
4365 
7189 

10-2090013 
2839 
5665 

8492 

10-2101319 
4148 
6977 
9808 

10-2n2639 
5471 
8304 

10-2121137 
3972 
6807 
9643 

10-2132480 
5318 
8156 

10-2140996 
3836 
6677 
9519 

10-2152362 

5206 
8051 

10-2160896 
3742 
6590 
9438 

10-2172287 
5136 
7987 

10-2180838 

3691 
6544 
9398 

10-2192253 
5109 
7966 

10-2200823 
3682 
6541 
9401 

10-2212263 
31° 



diff. 

2811 
2813 
2813 
2814 
2814 
2816 
2816 
2817 
2818 
2819 
2820 
2820 
2821 
2822 
2823 
2824 
2824 
2826 
2826 
2827 

2827 
2829 
2829 
2831 
2831 
2832 
2833 
2833 
2835 
2835 

2836 
2837 
2838 
2838 
2840 
2840 
2841 
2842 
2843 
2844 

2845 
2845 
2846 
2848 
2848 
2849 
2849 
2851 
2851 
2853 



2854 
2855 
2856 
2857 
2857 
2859 
2859 
2860 
2862 

diff. 



59° 

10-2212263 
5125 
7988 

10-2220851 
3716 
6582 
9448 

10-2232315 
5184 
8053 

10-2240923 
3794 
6666 
9538 

10-2252412 
5287 
8162 

10-2261039 
3916 
6794 

9673 
10-2272553 
5434 
8316 
10-2281199' 
4083 
6967 
9853 
10-2292739 
5627 

8515 
10-2301404 
4295 
7186 
10-2310078 
2971 
5865 
8760 
10-2321656 
4552 

7450 

10-2330349 
3249 
6149 
9051 

10-2341953 
4857 
7761 

10-2350666 
3573 
6480 
9388 

10-2362298 
5208 
8119 

10-2371031 
3944 
6858 
9773 

10-2382689 
5606 
30° 



diff 

2862 
2863 
2863 
2865 
2866 
2866 
2867 
2869 
2869 
2870 

2871 
2872 
2872 
2874 
2875 
2875 
2877 
2877 
2878 
2879 
2880 
2881 
2882 
2883 
2884 
2884 
2886 
2846 



2891 
2891 
2892 
2893 
2894 
2895 
2896 
2896 
2898 

2899 
2900 
2900 
2902 
2902 
2904 
2904 
2905 
2907 
2907 

2908 
2910 
2910 
2911 
2912 
2913 
2914 
2915 
2916 
2917 

diff 

LOG. COTAN. 



■ 60° 

10-2385606 
8524 

10-2391443 
4363 
7284 

10-2400206 
3129 
6053 
6978 

10-2411904 

4830 
7758 

10-2420687 
3617 
6548 
9480; 

10-2432413 
5347 
8282 

10-2441217 
4154 
7092 

10-2450031 
2971 
5912 
8854 

10-2461797 
4741 
7686 

10-2470632 

3580 
6528 
9477 

10-2482427 
5378 
8331 

10-2491284 
4238 
7194 

10-2500150 
3108 
6066 
9026 

10-2511987 
4948 
7911 

10-2520875 
3840 
6806 
9773 

10-2532741 
5710 
8680 

10-2541651 
4624 
7597 

10-2550572 
3547 
6524 
9501 

10-2562480 



2918 
2919 
2920 
2921 
2922 
2923 
2924 
2925 
2926 
2926 

2928 
2929 
2930 
2931 
2932 
2933 
2934 
2935 
2935 
2937 

2938 
2939 
2940 
2941 
2942 
2943 
2944 
2945 
2946 
2948 
2948 
2949 
2950 
2951 
2953 
2953 
2954 
2956 
2956 
2958 
2958 
2960 
2961 
2961 
2963 
2964 
2965 
2966 
2967 
2968 

2969 
2970 
2971 
2973 
2973 
2975 
2975 
2977 
2977 
2979 



61° 

10-2562480 
5460 
8441 

10-2571423 
4406 
7391 

10-2580376 
3362 
6350 
9338 

10-2592328 
5319 
8311 

10-2601304 
4298 
7293 

10-2610290 
3287 
6286 
9285 

10-2622286 
5288 
8291 

10-2631295 
4301 
7307 

10-2640315 
3323 
6333 
9344 

10 2652356 
5369 
8384 

10-2661399 
4416 
7434 

10-2670453 
3473 
6494 
9516 

10-2682540 
5564 
8590 

10-2691617 
4646 
7675 

10-2700705 
3737 
6770 
9804 

10-2712839 
5876 
8913 

10-2721952 
4992 
8033 

10-2731075 
4119 
7163 

10-2740209 
3256 
28° 



diff. 

2980 
2981 
2982 
2983 
2985 
2985 
2986 



2990 

2991 
2992 
2993 
2994 
2995 
2997 
2997 
2999 
2999 
3001 
3002 
3003 
3004 
3006 
3006 
3008 
3008 
3010 
3011 
3012 

3013 
3015 
3015 
3017 
3018 
3019 
3020 
3021 
3022 
3024 
3024 
3026 
3027 
3029 
3029 
3030 
3032 
3033 
3034 
3035 
3037 
3037 
3039 
3040 
3041 
3042 
3044 
3044 
3046 
3047 



118 



LOG. SINE. 



63° m- 64° dif. 
1-949 8S09 643 9-953 6602 .^g 

94d2^,.j (21^^.-,. 

1-950 0095 fil^ 75Q-.»i=' 

0738 g.^ 

1330 1% 



65° \dif. 66° 
•957 2757VeQ 9-960 7302 
3346 .^H 7864 



[TaWe II. 



620 

9-945 9349 
9-946 0021 
0692 
1362 
2032 
2702 
3371 
4040 
4708 
5376 

6043 
6710 
7376 
8042J 
8707 
9372 
9-947 0036 
0700 
1364 
2027 

2689 
3352 
4013 
4674 
5335 
5995 
6655 
7314 
7973 
8631 

9289 
9947 
9-943 0604 
1260 
1916 
2572 
3227 
3881 
4535 
5189 
5342 
6495 
7147 
7799 
8450 
9101 
9752 
0402 
1051 
1700 

2349 
2997 
3645 
4292 
4938 
5585 
6230 
6376 
7521 
8165 
8809 



9-949 



dif. 
672 
671 
670 
670 
670 
669 
669 
668 
668 
667 

667 
666 
666 
665 
665 
664 
664 
664 
663 
662 

663 
661 
661 
661 
660 
660 
659 
659 
658 
658 

658 
657 
656 
656 
656 
655 
654 
654 
654 
653 

653 
652 
652 
651 
651 
651 
650 
649 
649 
649 
648 
648 
647 
646 
647 
645 
646 
645 
644 
644 

dif. 



2022 ll: 9677 l\\ 

2663 g|i 9-954 0291 g^ 
3303 ^y 0904 ^lo 

4583 640 
5223.03 

5861639 
659^638 

Lli't 63: 

8412 637 

9049^36 

9b&o 63:; 

-951 03-20 g^g 

0956^.^^ 

1590 631 

999 1 5xt 



/o33 6ir 
8448 l\l 

9b/ < 614 



?9?4 6ii 
151' fiio 

2741 R„ 
3352 ^ 

4o/4 6iq9- 

5134 609 
5 '93 659 
6402 609 
1011608 
822. 607 
J834 607 



Y,y' 606 
06o3 606 
1259 60^^ 
1864 655 

2469^^4 
3073 603 

3676 nr^A 

4280^^2 
4882 603 9' 
^48^ 602 
60&/ 601 
6688 ^-^ 
7289 
7890 
8490 
9039 



601 
601 



600 
599 
600 



^^■^^^y 634 "-- 607 
2224 634 _„^ 9441606 

2SoS 634 9-950 004/ 606 
3492 cqo 06o3 cnc 

4124 63i 

4'^' 632 
53b963i 

602<3l 
^ooJ631 
^282630 

7912699 
8541 6i0 
9171 69Q 
9799 629 
1-952 0423 627 

1055 69q 

1683 6-i7 
2310 6-i6 

3562 626 

41S3'625 
4813624 
5437:694 
6061624 
6685 693 
7308 eis 
7931622 
8553 692 

9175'622 
97971621 

1-953 0413'620 
1038;690 
1658 620 
S^^619 
289.613 
Solo 619 
4134 617 

4/ol 6iq 
5369,616 
59S.O;6i7 
6602, .^ 



^-5 ^^6-26 y^'»y 600 
2936 6^6 9669 59S 
3562 6^6 9-956 0287 ^g 

4 188 '69.;^ 0386 597 
1433 593 
2031 I? 
2678 i^ 

^^^4 596 
3a /O 596 
44bfa XQ5 
5061 595 
5656 594 9 
6250 594 

6844 593 
7437 XQQ 
8030 593 
3623 59-5 
9215 591 
9306 59 
57 0397 591 
09SS 590 

IIB 539 



9-95' 



6284:586 
6870. ^Pfi 
7456!^86 

8041 S 



8626 
9210 



584 
534 



9794^4 

■958 0373 f^^ 
0961 f^l 

1543 rq9 

21-25 Too 

3288 °°; 
3869^81 

4450 5go 
5030 p§ 



6183 579 

/340 r-rc 

„q.-,„ / b 

p¥5r577 
90// 576 

9653 ^i^ 

)-959 02-29I576 
08051^7? 
1330 ^'^ 
1954 574 
2528 57I 



574 

4243^^^ 
4321 



dif 
i346|r^^"7S6a|2 60 
3934 1^ 8426^62 58 

4522,^°° 8937 if. 57 

5110,|^ 9548^61 ,g 

5697i^°| 9-961 0108 §60 55 
aooA.^^' 0663^60 ,^ 

1228.1^053 



F228| 53 
1787 5^5 52 
2346 fj 51 



50 
49 
48 
47 
46 



000 

2904 rrr. 

3462 11° 

4020 5^6 
4o/b557 

5133556 46 

5689 r rfi 45 

6245 I55 44 

6800 „r 43 

7355 ^K4 42 

7909 %\ 41 

8463'553 40 

0U3U 570 9016 Kr.> 39 

5609 570 9569 r„ 38 

6183 570 9-962 0122^^5 37 

SI 578 ?S! 55^36 



573 
5393571 
5964 571 



6535 57J 9994i^|^ 

-''^^^7^9-963 05331^^1 



2757^.. 
250 dif 

LOG. COSINE. 



1106 570 
'6'6{570 

824656 

68l0r6D 
9902:-;/^q 

'■S™ 05201^1 

4434 564 
5043 561 
5612 i64 
61^6 563 



0674 552 ^'3 
1226551 35 

1777 rri 34 I 

2323j??n 33 ^ 

^n 

3978'54Q 30 ' 

4527 e^^ 29 ! 

5076'?4Q 28 

56-24 ^;!q 27 

6172112 26 

6719.5I; 25 

7266,^1^ 24 

7812,r4fi 23 

8353||l6 22 
8904 e46 21 

9449I545 20 
99941^4^ 19 
ncQQi^'*'* ig 



o^"^563 y^°' 537 

6739 563 9724 ^^7 

7302^^^^ 9-964 026rl^ 
240 dif 230 ^'/ 



— 543 17 

I62515I3 16 

2168:513 15 

27II542 14 

3253II2 13 

3795:^lf ^2 

4336|li 11 

5417112 9 

'«^S!539 ? 

^n^fi540 I 

:t2^5i538 I 

7574roQ 5 

81125! 4 

8650 53? 3 

9187 ll 2 

512453? J 



i: 



TaUe II.] 



LOG. TAN. 



119 



62^ 

10-2743256 
46305 
49354 
52405 
55457 
58510 
61564 
64619 
67676 
70734 

73793 
76853 
79915 
82978 
86042 
89107 
92173 
95241 
98310 
10-2801380 

04451 
07524 
10598 
13673 
16749 
19827 
22906 
25986 
29067 
32149 
35233 
38318 
41405 
44492 
47581 
50671 
53763 
56855 
59949 
63044 

66141 
69239 
72338 
r5438 
78539 
81642 
84746 
87852 
90959 
94067 
97176 
10-2900287 
03399 
06512 
09626 
12742 
15859 
18978 
22098 
25219 
28341 
27' 



3049 
3049 
3051 
3052 
3053 
3054 
3055 
3057 
3058 
3059 

3060 
3062 
3063 
3064 
3065 
3066 
3068 
3069 
3070 
3071 
3073 
3074 
3075 
3076 
3078 
3079 
3080 
3081 
3082 
3084 

3085 
3087 
3087 
3089 
3090 
3092 
3092 
3094 
3095 
3097 

3098 
3099 
3100 
3101 
3103 
3104 
3106 
3107 
3108 
3109 

3111 
3112 
3113 
3114 
3116 
3117 
3119 
3120 
3121 
3122 

\diff. 



63° 
10-2928341 
31465 
34590 
37716 
40844 
43973 
47103 
50235 
53368 
56503 
59638 
62775 
65914 
69054 
72195 
75337 
78481 
81626 
84773 
87920 

91070 
94220 
97372 
10-3000526 
03680 
06836 
09994 
13153 
16313 
19474 

22637 
25802 
28968 
32135 
35303 
38473 
41645 
44817 
47991 
51167 

54344 
57522 
60702 
63883 
67066 
70250 
73435 
76622 
79811 
83000 

86191 
89384 
92578 
95774 
98970 
10-3102169 
05369 
08570 
11773 
14977 
18182 
26° 



3124 
3125 
3126 
3128 
3129 
3130 
3132 
3133 
3135 
3135 
3137 
3139 
3140 
3141 
3142 
3144 
3145 
3147 
3147 
3150 

3150 
3152 
3154 
3154 
3156 
3158 
3159 
3160 
3161 
3163 
3165 
3166 
3167 
3168 
3170 
3172 
3172 
3174 
3176 
3177 

3178 
3180 
3181 
3183 
3184 
3185 
3187 
3189 
3189 
3191 
3193 
3194 
3196 
3196 
3199 
3200 
3201 
3203 
3204 
3205 



64° 

10-3118182 
21389 
24598 
27808 
31019 
34232 
37447 
40662 
43880 
47099 



50319 
53541 
66764 
59989 
63215 
66443 
69672 
72902 
76135 
79368 
82604 
85840 
89079 
92318 
95560 
98802 
10-3202047 
05292 
08540 
11789 

15039 
18291 
21544 
24799 
28056 
31314 
34574 
37835 
41097 
44362 
47628 
50895 
54164 
57434 
60706 
63980 
67255 
70532 
73810 
77090 
80372 
83655 
86940 
90226 
93514 
96803 
10-3300094 
03387 
06681 
09977 noQO 
13275^"^^^ 
250 Uiff. 

LOG. COTAN. 



diff. 

3207 

3209 

3210 

3211 

3213 

3215 

3215 

3218 

3219 

3220 

3222 

3223 

3225 

3226 

3228 

3229 

3230 

3233 

3233 

3236 

3236 

3239 

3239 

3242 

3242 

3245 

3245 

3248 

3249 

3250 

3252 

3253 

3255 

3257 

3258 

3260 

3261 

3262 

3265 

3266 

3267 

3269 

3270 

3272 

3274 

3275 

3277 

3278 

3280 

3282 



3283 
3285 
3286 
3288 
3289 
3291 
3293 
3294 
3296 



65° 

10-3313275 
16574 
19874 
23177 
26481 
29786 
33093 
36402 
39712 
43025 

46338 
49654 
52970 
56269 
59609 
62931 
66255 
69580 
72907 
76235 

79566 
82897 
86231 
89566 
92903 
96242 
99582 
10-3402924 
06267 
09613 

12959 
16308 
19659 
23011 
26364 
29720 
33077 
36436 
39796 
43159 
46523 



53256 
56625 
59996 
63369 
66743 
70119 
73497 
76877 

80258 
83641 
87026 
90413 
93801 
97191 
10-3500583 
03977 
07372 
10770 
14169 
240 



3299 
3300 
3303 
3304 
3305 
3307 
3309 
3310 
3313 
3313 

3316 
3316 
3319 
3320 
3322 
3324 
3325 
3327 
3328 
3331 
3331 
3334 
3335 
3337 
3339 
3340 
3342 
3343 
3346 
3346 

3349 
3351 
3352 
3353 
3356 
3357 
3359 
3360 
3363 
3364 

3365 
3368 
3369 
3371 
3373 
3374 
3376 
3378 
3380 
3381 

3383 
3385 
3387 
3388 
3390 
3392 
3394 
3395 
3398 
3399 



66° 

10-3514169 
17569 
20972 
24376 
27783 
31190 
34600 
38012 
41425 
44840 
48257 
51676 
55097 
58519 
61943 
65369 
68797 
72227 
75658 
79092 
82527 
85964 
89403 
92844 
96286 
99731 

10-3603177 
06625 
10075 
13527 

16981 
20437 
23894 
27354 
30815 
34278 
37743 
41210 
44679 
48150 

51622 
55097 
58574 
62052 
65532 
69015 
72499 
75985 
79473 
82963 

86455 



93444 
96942 
10-3700442 
03943 
07447 
10952 
14460 
17969 
21481 
23° 



diff. 

3400 
3403 
3404 
3407 
3407 
3410 
3412 
3413 
3415 
3417 

3419 
3421 
3422 
3424 
3426 
3428 
3430 
3431 
3434 
3435 
3437 
3439 
3441 
3442 
3445 
3446 
3448 
3450 
3452 
3454 

3456 
3457 
3460 
3461 
3463 
3465 
3467 
3469 
3471 
3472 

3475 
3477 
3478 
3480 
3483 
3484 
3486 
3488 
3490 
3492 

3493 
3496 
3498 
3500 
3501 
3504 
3505 
3508 
3509 
3512 



120 



[Table 





1 
2 

I 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 
23 

24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



67^ 
3-964 0261 
0797 
1332 

1868 
2402 
293' 
3470 
4004 
4537 
5069 

5602 
6133 
666 
7195 
7726 
8256 
87S5 
9314 
9843 
1-965 0371 

0899 
1426 
1953 
2480 
3006 
3532 
4057 
4582 
5106 
5630 
6153 
6677 
7199 
7721 
8243 
8764 
9285 
9S06 
■966 0326 
0346 

1365 
1834 
2402 
2920 
3437 
3954 
4471 
4987 
5503 
601S 
6533 
7048 
7562 
8075 
8583 
9101 
9614 
•967 0125 
0637 
1148 
1659 
22° 



536 
535 
536 
534 
535 
533 
534 
533 
5321 
533 

531 

532 

530 

531 

530 

529 

529 

529 

528 

528 

527 

527 

52 

526 

526 

525 

525 

524 

524 

523 

524 

22 
522 

22 
521 
521 
521 
520 
520 
519 

19 
518 
518 
517 
517 
517 
516 
516 
515 
515 

515 
514 
513 
513 
513 
513 
511 
512 
511 
511 



68° 
9-967 1659 
2169 



2679 
3188 
3697 
4205 
4713 
5221 
5728 
6235 
6741 
724 
7753 
825 
8763 
9267 
97 
9-963 0274 
077 



dij 



dif.\ 



1279 
1781 
2283 
2784 
3285 
3786 
4236 
4785 
5284 
5733 
6-281 

6779 
7276 
7773 
8270 
8766 
9262 
975 
9-969 0-252 
0746 
1241 
1734 
2227 
2720 
3212 
3704 
4196 
4687 
5177 
5668 
6158 

6647 
7136 
7624 
8112 
8600 
9087 
9574 
9-970 0061 
0547 
1032 
151 
21° 



510 
510 
509 
509 

508 

508 

503 

507 

507 

506 

506 

506 

505 

505 

504 

504 

503 

503 

502 

502 

502 

501 

501 

501 

500 

499 

4991 

499 

493 

498 

497 

497 

49' 

496 

496 

495 

495 

494 

495 

493 

493 
493 
492 
492 
492 
491 
490 
491 
490 
189 

489 
483 
433 
433 
437 
437 
487 
436 
485 
485 

dif: 



69° 

970 1517 
2002 
2486 
2970 
3454 
3937 
4419 
490'^ 
5383 
5865 
6346 
6826 
7306 
7786 
8265 
8744 
9223 
9701 
9-971 0173 
0655 
1132 
16US 
2034 
2560 
3035 
3509 
3934 
4457 
4931 



dif. 

485 
484 
484 
434 
483 
482 
483 
431 
482 
481 

480 

430 

460 

479 

4791 

479, 

4781 

477 

477, 

477 

476! 

476! 

476; 

475 

474 

475 

473 

474 

473 



70° 

9-972 9i 

9-973 0318 
0777 
1236 
1694 
2152 
2610 
3067 
3523 
3980 
4435 
4891 
5346 
5801 
6255 
6709 
7162 
7615 



5404.,:2 
5376479 
""'"472 
471 
471 
471 
470 
469 
470 
468 



6348: 
6320 
7291 
7762 
8233 
8703 
9172 
9642 
9-972 0110 

0579 
1047 
1514 
1931 
2448 
2914 
3330 
3845 
4310 
4775 

5239 
5703 
6166 
6629 
7092 
7554 
8016 
8477 
8938 
9398 
9358 

LOG. COSINE. 



8519 
8971 
9422 
9873 
9-974 0324 
0774 
1224 
1673 
2122 
2570 
3018 

3466 
3913 
4359 
4806 
5252 
5697 
6142 
6587 
7031 
7475 
7918 
8361 
8804 
9246 
9683 
9-975 0129 
0570 
1011 
1451 
1891 
2330 
2769 
3208 
3646 
4033 
4521 
4957 
5394 
5830 
6265 
6701 



19^ 



dif. 

460 

459 

459 

458 

458 

458 

457 

456 

457 

455 

456 

455 

455 

454 

454 

453 

453 

452 

452 

452 

451 

451 

451 

450 

450 

449 

449 

448 

448 

448 

447 

446 

447 

446 

445 

445 

445 

444 

444 

443 

443 

443 

442 

442 

441 

441 

441 

440 

440 

439, 

439 

439 

438 

437 

438 

436 

437 

436 

435 

436 



9-9^ 



10 ]dif 
5 6701 



7135 
7570 
8004 
843 
6870 
9303 
9736 
9-976 0167 
0599 
1030 
1461 
1891 
2321 
2750 
3179 
3608 
4036 
4464 
4391 
5318 
5745 
6171 
6597 
7022 
7447 
7872 
8296 
8720 
9143 

9566 

9983 

9-977 0410 

0832 



dif 



1253 
1674 
2095 
2515 
2934 
3354 

3772 
4191 
4609 
5026 
5444 
5860 
6277 
6693 
7103 
7523 

7938 
8353 
8766 
9180 
9593 
9-978 0006 
0418 
0830 
1241 
1653 
2063 
18° 



434 

435 

434 

413 

433 

433 

433 

431 

432 

431 

431 

430 

430 

4-29 

429 

429 

4-28 

4-28 

427 

427 

427 

426 

426 

425 

4'25 

425 

424 

424 

423 

423 

422 

422 

422 

421 

421 

421 

420 

419 

420 

418 

419 

418 

417 

418 

416 

417 

416 

415 

415 

415 

415 

413 

414 

413 

413 

412 

412 

411 

412 

410 

dif 



60 
59 

58' 
57 1 
56 I 
55 
54, 
53 
52 I 

51 : 

50' 
49 

43' 

47 

46; 

45: 

44, 

43, 

42; 

41 I 

40 I 

39 j 

38 

37 1 

36 i 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 
19 

18 
17 
16 
15 

14 
13 

12 
11 
10 

9I 
8| 
7 

el 

5' 

4 

3 

2 

1 





I Table ii.J 



LOG. TAN. 



121 



10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 



67^ 
iO-37214Sl 
24994 
2S509 
3202 
35546 
39068 
42591 
46U6 
49644 
53173 
56704 
60237 
63773 
67310 
70850 
74391 
77934 
81480 
85027 
88577 
92128 
95682 
99238 
10-3S02795 
06355 
09917 
13481 
17047 
20615 
241S5 
27757 
31331 
34907 
38486 
42066 
45649 
49234 
52820 
56409 
60000 
63593 
67188 
70786 
74385 
77987 
81591 
85196 



diff. 

3513 
3515 
3518 
3519 
3522 
3523 
3525 
3528 
3529 
3531 

3533 
3536 
3537 
3540 
3541 
3543 
3546 
3547 
3550 
3551 



92414 
96027 

99641 
10-3903258 
06876 
10497 
14120 
17746 
21373 
25003 
28634 
32268 
35904 
22° 



3554 
3556 
3557 
3560 
3562 
3564 
3566 
3568 
3570 
3572 
3574 
3576 
3579 
3580 
3583 
3585 
3586 
3589 
3591 
3593 

3595 
3598 
3599 
3602 
3604 
3605 
3608 
3610 
3613 
3614 

3617 
3618 
3621 
3623 
3626 
3627 
3630 
3631 
3634 
3636 

dvff. 



10-3935904 
39543 
43183 
46826 
50471 
54118 
57767 
61419 
65073 
68729 
72387 
76047 
79710 
83375 
87042 
90711 
94383 
98057 
0-4001733 
05412 
09092 
12775 
16460 
20148 
23838 
27530 
31224 
34921 
38620 
42321 
46025 
49731 
53439 
57149 
60862 
64577 
68295 
72015 
75737 
79461 
83188 
86918 
90649 
94383 
98119 
10-4101858 
05599 
09343 
13088 
16837 

20587 
24340 
28096 
31853 
35614 
39376 
43141 
46909 
50679 
54451 
58226 
21° 



diff. 

3639 
3640 
3643 
3645 
3647 
3649 
3652 
3654 
3656 
3658 
3660 
3663 
3665 
3667 
3669 
3672 
3674 
3676 
3679 



3683 
3685 
3688 
3690 
3692 
3694 
3697 
3699 
3701 
3704 
3706 
3708 
3710 
3713 
3715 
3718 
3720 
3722 
3724 
3727 
3730 
3731 
3734 
3736 
3739 
3741 
3744 
3745 
3749 
3750 

3753 
3756 

3757 
3761 
3762 
3765 
3768 
3770 
3772 
3775 



69° 

10-4158226 
62003 
65783 
69565 
73349 
77136 
80926 
84718 
88512 
92309 
96108 
99910 

10-4203714 
07521 
11331 
15142 
18957 
22774 
26593 
30415 
34239 
38066 
41896 
45728 
49562 
53399 
57239 
61081 
64926 
68773 
72623 
76476 
80331 
84189 



3777 

3780 
3782 
3784 
3787 
3790 
3792 
3794 
3797 
3799 
3802 
3804 
3807 
3810 
3811 
3815 
3817 
3819 
3822 
3824 



91912 
95777 
99645 
LO-4303516 
07389 
11265 
15144 
19025 
22909 
26795 
30684 
34576 
38470 
42367 
46267 

50169 
54075 
57982 
61893 
65806 
69722 
73640 
77561 
81485 
85412 
89341 
20° 

LOG. COTAN. 



3827 
3830 
3832 
3834 
3837 
3840 
3842 
3845 
3847 
3850 
3853 
3855 
3858 
3860 
3863 
3865 



3871 
3873 
3876 

3879 

3881 
3884 



3892 
3894 
3897 
3900 
3902 

3906 
3907- 
3911 
3913 
3916 
3918 
3921 
3924 
3927 
3929 



70° 

10-4389341 
93273 
9720& 

10 4401146 
05086 
09029 
12975 
16923 
20875 
24829 
28786 
32745 
36708 
40673 
44641 
48612 
52585 
56562 
60541 
64523 
68508 
72496 
76486 
80479 
84476 
88475 
92477 
96481 
10-4500489 
04500 
08513 
12529 
16548 
20570 
24595 
28623 
32654 
36688 
40724 
44764 
48807 
52852 
56900 
60952 
65006 
69063 
73123 
77187 
81253 
85322 



diff 
3932 
3935 
3938 
3940 
3943 
3946 
3948 
3952 
3954 
3957 

3959 
3963 
3965 
3968 
3971 
3973 
3977 
3979 
3982 



89394 
93469 
97547 
10 4601629 
05713 
09800 
13890 
17983 
22080 
26179 
30281 
19« 



3990 
3993 
3997 
3999 
4002 
4004 
4008 
4011 
4013 
4016 
4019 
4022 
4025 
4028 
4031 
4034 
4036 
4040 
4043 

4045 
4048 
4052 
4054 
4057 
4060 
4064 
4066 
4069 
4072 

4075 
4078 
4082 
4084 
4087 
4090 
4093 
4097 
4099 
4102 

diff 



71° 

10-4630281 
34387 
38495 
42607 
46722 
50839 
54860 
59084 
63211 
67341 

7U7A 
75611 
79750 
83893 
88039 
92187 
96339 
10 4700495 
04653 
08814 

12979 
17147 
21318 
25492 
29669 
33850 
38034 
42221 
46411 
50605 
54801 
59001 
63205 
67411 
71621 
75834 
80050 
84270 
88492 
92718 
96948 
10-4801181 
05417 
09656' 
13899 
18145 
22394 
26647 
30903 
35162 

39425 
43691 
47961 
52234 
56510 
60790 
65073 
69359 
73649 
77943 
82240 
18° 



4106 
4108 
4112 
4115 
4117 
4121 
4124 
4127 
4130 
4133 
4137 
4139 
4143 
4146 
4146 
4152 
4156 
4158 
4161 
4165 
4168 
4171 
4174 
4177 
4181 
4184 
4187 
4190 
4194 
4196 
4200 
4204 
4206 
4210 
4213 
4216 
4220 
4222 
4226 
4230 
4233 
4236 
4239 
4243 
4246 
4249 
4253 
4256 
4259 
4263 

4266 
4270 
4273 
4276 
4280 
4283 
4286 
4290 
4294 
4297 



dtff. 



}h 



122 



LOG. SINE. 



[Table II. 



10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



72° 
•978 2063 
2474 
2883 
3293 
3702 
4111 
4519 
4927 
5334 
5741 

6148 
6554 
6960 
7365 
7770 
8175 
8579 
8983 
9386 
9789 

•979 0192 
0594 
0996 
1397 
1798 
2198 
2599 
2998 
3398 
3796 
4195 
4593 
4991 
5388 
5785 
6182 
6578 
6973 
7369 
7764 
8158 
8552 
8946 
9339 
9732 

•980 0124 
0516 
0908 
1299 
1690 

2081 
2471 
2860 
3250 
3639 
4027 
4415 
4803 
5190 
5577 
5963 
170 



dif. 



9-980 5963 
6349 
6735 
7120 
7505 
7889 
8273 
8657 
9040 
9423 
9805 

9-981 0187 
0569 
0950 
1331 
1711 
2091 
2471 
2850 
3229 
3608 



4363 
4740 
5117 
5494 
5870 
6245 
6620 
6995 
7370 
7744 
811 
8490 
8863 
9236 
9608 
9979 
9-982 0351 
0721 
1092 
1482 
1831 
2201 
2569 
2938 
3308 
3674 
4041 
4408 

4774 
5140 
5506 
5871 
6236 

6600|gr^ 
69641.364 
732Q1364 

7691 
8054 
8416 
160 



dif. 

386 
386 
385 
385 
384 
384 
384 
383 
383 
382 
382 
382 
381 
381 
380 
380 
380 
379 
379 
379 
378 
377 
377 
377 
377 
376 
375 
375 
375 
375 
374 
373 
373 
373 
373 
372 
371 
372 
370 
371 
370 
369 
370 
368 
369 
368 
368 
367 
367 
366 
366 
366 
365 
365 
1364 



363 
!363 
362 



W- 



dif. 

362 
362 
361 
361 
361 
360 
359 
360 
■359 
358 
358 
358 
357 
357 
356 
356 
356 
355 
355 
355 
354 
354 
353 
353 
352 
353 
351 
352 
351 
350 
350 
350 
349 
349 
349 
348 
348 
347 
347 
347 
346 
346 
345 
345 
345 
344 
344 
343 
343 
343 
342 
342 
342 
341 
340 
341 
339 
340 
339 
339 

dif 

LOG. COSINE. 



74° 

9-982 8416 
8778 
9140 
9501 
9862 

9-983 0223 
0583 
0942 
1302 
1661 

2019 
2377 
2735 
3092 
3449 
3605 
416] 
4517 
4872 
5227 
5582 
5936 
6290 
6643 
6996 
7348 
7701 
8052 
8404 
8755 

9105 
9455 
9805 
9-984 0154 
0503 
0852 
1200 
1548 
1895 
2242 
2589 
2935 
3281 
3626 
3971 
4316 
4660 
5004 
5347 
5690 

60331 
6375 
6717 
7059 
7400 
7740 
8081 
8420 
8760 
9099 
9438 
15° 



75° 

•984 9438 
9776 

-985 0114 
0452 
0789 
1125 
1462 
1798 
2133 
2468 
2803 
3138 
3471 
3805 
4138 
4471 
4803 
5135 
5467 
5798 

6129 
6460 
6790 
7119 
7449 
7777 
8106 
8434 
8762 
9089 
9416 
9742 
1-986 0069 
0394 
072i 
1045 
1369 
1693 
2017 
2340 
2663 
2986 
3308 
3630 
3952 
4273 
4593 
4913 
5233 
5553 
5872 
6191 
6509 
6827 
7144 
7461 
7778 
8094 
8410 
8726 
9041 
14° 



dif 

338 
338 
338 
337 
336 
337 
336 
335 
335 
335 
335 
333 
334 
333 
333 
332 
332 
332 
331 
331 
331 
330 
3-29 
330 
328 
329 
328 
3-28 
327 
327 
3-26 
327 
325 
3-26 
325 
324 
324 
324 
323 
323 
323 
322 
322 
322 
321 
320 
320 
320 
320 
319 
319 
318 
318 
317 
317 
317 
316 
316 
316 
315 

dif] 



76° 

-986 9041 
9356 
9670 
9984 

-987 0298 
0611 
0924 
1236 
1549 
1860 
2171 
2482 
2793 
3103 
3413 
3722 
4031 
4339 
4648 
4955 
5263 
5570 
5876 
6183 
6488 
6794 
7099 
7404 
7708 
8012 
8315 
8618 
8921 
9223 
9525 
9827 

1-988 0128 
0429 
0729 
1029 
1329 
1628 
1927 
2225 
2523 
2821 
3118 
3415 
3712 
4008 
4303 
4599 
4894 
5188 
5482 
5776 
6070 
6363 
6655 
6947 
7239 
13° 



dif 

315 
314 
314 
314 
313 
313 
312 
313 
311 
311 
311 
311 
310 
310 
309 
309 
308 
309 
307 
SOS 
307 
306 
307 
305 
306 
305 
305 
304 
304 
303 
303 
303 
302 
302 
302 
301 
301 
30G 
300 
300 
299 
299 
298 
•298 
298 
297 
297 
297 
296 
295 
296 
295 
294 
294 
294 
294 
•293 
292 
292 
292 

dif 



\Table ii.] 



123 



10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 

34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



7-2=^ \di 
10-4SS2240 
86540 
90844 
95151 
99461 
10-4903776 
08093 
12414 
16739 
21067 



25398 
29733 
34072 
38414 
42760 
47109 
51462 
55818 
60178 
64541 

68908 
73279 
77653 
82031 
86412 
90797 
95186 
99578 
10-5003974 
08374 
12777 
17184 
21594 
26009 
30426 
34848 
39273 
43702 
48135 
52571 
57012 
61455 
65903 
70354 
74810 
79269 
83731 
88198 
92668 
97142 

10-5101620 
06102 
10587 
15076 
19570 
24067 
28567 
33072 
37581 
42093 
46610 
170 



4300 
4304 
4307 
4310 
4315 
4317 
4321 
4325 
4328 
4331 



4335 

4339 
4342 
4346 
4349 
4353 
4356 
4360 
4363 
4367 

4371 
4374 
4378 
4381 
4385 
4389 
4392 
4396 
4400 
4403 

4407 
4410 
4415 
4417 
4422 
4425 
4429 
4433 
4436 
4441 

4443 
4448 
4451 
4456 
4459 
4462 
4467 
4470 
4474 
4478 
4482 
4485 
4489 
4494 
4497 
4500 
4505 
4509 
4512 
4517 



73° \dijf 
10-5146610 
51130 
55654 
60182 
64714 
69250 
73790 
78334 
82882 
87434 

91989 
96549 
10 5-201113 
05681 
10252 
14828 
19408 
23991 
28579 
33171 

37767 
42367 
46971 
51579 
56192 



65428 
70053 
74682 
79315 
83952 
88593 
93238 
97888 

10-5302541 
07199 
11861 
16527 
21198 
25873 
30552 
35235 
39922 
44614 
49310 
54010 
58715 
63424 
68137 
72855 
77577 
82303 
87033 
91768 
96508 

10-5401251 
05999 
10752 
15509 
20270 
25036 
16° 



4520 
4524 
4528 
4532 
4536 
4540 
4544 
4548 
4552 
4555 

4560 
4564 
4568 
4571 
4576 
4580 
4583 



4592 
4596 

4600 
4604 
4608 
4613 
4616 
4620 
4626 
4629 
4633 
4637 

4641 
4645 
4650 
4653 
4658 
4662 
4666 
4671 
4675 
4679 
4683 
4687 
4692 
4696 
4700 
4705 
4709 
4713 
4718 
4722 

4726 
4730 
4735 
4740 
4743 
4748 
4753 
4757 
4761 
4766 



740 
10-5425036 
29806 
34580 
39359 
44143 
48931 
53724 
58521 
63322 
68128 

72939 

77754 
82573 
87398 
92226 
97060 

10-5501898 
06740 
11587 
16439 
21296 
26157 
31022 
35893 
40768 
45648 
50532 
55421 
60315 
65214 
7011^ 
75025 
79938 
84855 
89778 
94705 
99637 

10-5604574 
09515 
14462 



diff. 

4770 
4774 
4779 
4784 
4788 
4793 
4797 
4801 
4806 
4811 
4815 
4819 
4825 
4828 
4834 
4838 
4842 
4847 
4852 
4857 
4861 
4865 
4871 
4875 
4880 
4884 



19413 
24369 
29330 
34296 
39267 
44243 
49224 
54209 
59200 
64195 



69196 
74201 
79211 
84227 
89247 
94273 
99303 
10-5704339 
09379 
14425 
19475 
150 

LOG. COTAN 



4894 
4899 
4903 

4908 
4913 
4917 
4923 
4927 
4932 
4937 
4941 
4947 
4951 
4956 
4961 
4966 
4971 
4976 
4981 
4985 
4991 
4995 
5001 
5005 
5010 
5016 
5020 
5026 
5030 
5036 
5040 
5046 
5050 



750 
10-5719475 
24531 
29592 
34658 
39725 
44806 
49887 
54974 
60065 
65162 

70265 
75372 
80485 
85602 
90725 
95854 
10-5800987 
06126 
11271 
16420 

21575 
26735 
31901 
37072 
42248 
47430 
52617 
57809 
63007 
68211 

73419 
78634 
83854 
89079 
94310 
99546 
10-5904788 
10035 
15288 
20547 

25811 
31081 
36356 
41637 
46924 
52216 
57514 
62818 
68127 
73442 

78763 
84090 
89422 
94760 
10-6000104 
05453 
10809 
16170 
21537 
26911 
32289 
140 



diff. 

5056 
5061 
5066 
5071 
5077 
5081 
5087 
5091 
5097 
5103 
5107 
5113 
5117 
5123 
5129 
5133 
5139 
5145 
5149 
5155 

5160 
5166 
5171 
5176 
5182 
5187 
5192 
5198 
5204 
5208 
5215 
5220 
5225 
5231 
5236 
5242 
5247 
5253 
5259 
5264 

5270 
5275 
5281 
5287 
5292 
5298 
5304 
5309 
5315 
5321 
5327 
5332 
5338 
5344 
5349 
5356 
5361 
5367 
5374 
5378 

diff 



76° 
10-6032289 
37674 
43065 
48462 
53864 
59273 
64687 
70107 
75534 
80966 



86405 
91849 
97300 
10-6102756 
08219 
13688 
19163 
24644 
30131 
35624 

41124 
46630 
52142 
57660 
63184 
68715 
74252 
79795 
85345 
90900 
96463 
10-6202031 
07606 
13187 
18775 
24369 
29970 
35577 
41190 
46810 

52437 
58070 
63709 
69355 
75008 
80667 
86333 
92006 
97685 
10-6303371 
09063 
14762 
20468 
261.81 
31900 
37626 
43359 
49099 
54845 
60599 
66359 
130 



diff 

5385 
5391 
5397 
5402 
5409 
5414 
5420 
5427 
5432 
5439 
5444 
5451 
5456 
5463 
5469 
5475 
5481 
5487 
5493 
5500 
5506 
5512 
5518 
5524 
5531 
5537 
5543 
5550 
5555 
5563 
5568 
5575 
5581 
5588 
5594 
5601 
5607 
5613 
5620 
5627 
5633 
5639 
5646 
5653 
5659 
5666 
5673 
5679 
5686 
5692 

699 
5706 
5713 
5719 
5726 
5733 
5740 
5746 
5754 
5760 



diff 



124 



LOG. SINE. 



[Table II. 



11° \dif. 
9-9887 239L„., 

531 292 

822 
9-9B88 113 

403 

693 

982 
9-9889 271 

560 

849 

9-9890 137 

424 

711 

998 
9-9891 285 

571 

856 
9-9392 142 

427 

711 



995 
279 
562 
845 

9-9894 126 
410 
692 
973 

9-9895 254 
535 
815 

9-9896 095 
374 
654 
932 

9-9897 211 
489 
766 

9-9898 043 
320 
597 
873 

9-9899 148 
4-23 
698 
973 

9-9900 247 
521 
794 

9-9301 06 

33? 
612 
883 

9-9902 155 
42e 
697 
967 

9-9903 237 
506 

9-9904 044 
120 



289 
289 
288 
287 
■287 
287 
287 
286 
285 
286 
285 
284 
284 
284 
283 
283 
283 
282 
282 
281 
281 
281 
280 
280 
279 
280 
278 
279 
278 
277 
277 
277 
•277 
276 
275 
275 
275 
275 
274 
274 
273 
•273 
272 

273 
•271 
•272 
•271 
■271 
270 
270 
269 
269 
269 

dif. 



78° Idif. 
9-9904 044Lo 

565'^^° 

848 
9-9905 115 

382 

648 

914 
9-9906 180 

445 

710 

974 
9-9907 239 

502 

766 
9-9908 029 

291 

553 

815 
9-9909 077 

338 

598 

859 
9-9910 119 

378 

637 

896 
9-9911 154 

412 

670 

927 
9-9912 184 

440 

696 

952 
9-9913 207 

462 

717 

971 
9-9914 225 

478 

731 

984 

9-9915 236 



739 
990 
9-9916 241 
492 
741 

991 

9-9917 240 

489 

737 

988 

9-9918 233 

480 

727 

974 

9-9919 220 

466 

11° 



268 
267 
267 
266 
266 
266 
265 
265 
■264 
265 
263 
264 
■263 
262 
262 
262 
262 
261 
260 
261 
260 
259 
259 
259 
258 
258 
258 
257 
257 
256 
256 
•256 
255 
■255 
255 
254 
254 
253 
253 
253 
252 
252 
251 
251 
251 
251 
•249 
250 

249 
249 
248 
249 
247 
247 
•247 
247 
246 
246 

dif. 



79° 

9-9919 466 
711 
956 

9-9920 201 
445 
689 
932 

9-9921 175 
418 
660 
902 

9-9922 144 
385 
626 



9-9923 106 
346 
585 
824 

9-9924 063 
301 
539 
776 

9-9925 013 
250 
486 
722 
957 

9-9926 192 
427 
661 
895 

9-9927 129 
362 
59 
827 

9-9928 059 
291 
522 
753 
984 

9-9929 214 
444 
673 
902 

9-9930 131 
359 
587 
814 

9-9931 041 

268 
494 
720 
946 

9-9932 171 
396 
621 
845 

9-9933 068 
292 
515 
10° 



dif. 

245 

245 

•245 

244 

•244 

243 

243 

243 

•242 

•242 

242 

241 

241 

240 

240 

240 

239 

239 

239 

238 

238 

237 

237 

•237 

236 

236 

235 

235 

235 

234 

234 

234 

233 

233 

232 

232 

232 

231 

•231 

231 

230 

230 

229 

2^29 

229 

228 

228 

2^27 

227 

227 

226 

226 

•226 

225 

225 

225 

224 

223 

224 

223 



80° \dif. 
9-9933 515I222 

959I222 

9-9934 18l|222 

403222 

624 221 

844 



81° 
9-9946 199 
399 



diff. 



9-9935 065 

285 
504 

723 
942 

9-9936 160 
378 
596 
813 

9-9937 030 
247 
463 
679 
894 

9-9938 109 
324 
538 
752 
965 

9-9939 178 
391 
603 
815 

9-9940 027 
238 
449 
659 
870 

9-9941 07£ 
289 
496 
706 
914 

9-9942 122 
330 
537 
743 
950 

9-9943 156 
361 
566 
771 
975 

9-9944 180 
383 
56r7 
789 
992 

9-9945 194 
396 
597 
798 
999 

9-9946 199 
9° 



220 
221 
220 
219 
19 
219 
218 
218 
218 
217 
217 
217 
216 
216 
215 
215 
215 
214 
214 
•213 
213 
213 
212 

m 

212 
211 
•211 
210 
•211 
209 
210 
209 
208 
208 
208 
208 
207 
•206 
207 
206 
205 
•205 
205 
204 
205 

•203 
•204 
•202 
203 
202 
202 
•201 
•201 
'201 
200 



dif. 

200 

599 200 



798 
997 
9-9947 195 
393 
591 
788 
985 

9-9948 181 
377 
573 
769 
964 

9-9949 158 
352 
546 
740 
933 

9-9950 126 
316 
510 
702 
893 

9-9951 084 
274 
464 
654 
844 

9-9952 033 
221 
409 
597 
785 
972 

9-9953 159 
345 
531 
717 
902 

9-9954 087 

271 

45 

639 

822 

9-9955 005 
188 
370 
552 

734 
915 

9-9956 095 
276 
456 
635 
815 
993 

9-9957 172 
350 
528 



199 
199 
198 
198 
198 
197 
197 
196 
196 
196 
196 
195 
194 
194 
194 
194 
193 
193 
192 
192 
192 
191 
191 
190 
190 
190 
90 
89 
188 
188 
188 
188 
187 
187 
186 
186 
186 
185 
185 
184 
184 
164 
183 
183 
183 
182 
182 
182 
181 
180 
181 
180 
179 
180 
178 
179 
178 
178 



60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

-20 

19 

18 

17 

16 

15 

14 

13 

12 

11- 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 





LOG. COSINE. 



Table ii.] 



LOG. TAN. 



125 



11^ 
010-6366359 



9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
I54 
155 
56 
57 
58 
59 
60 



72126 
77900 
83681 
89469 
95264 

10-6401065 
06874 
12690 
18513 
24342 
30179 
36023 
41874 
47733 
53598 
59470 
65350 
71237 
77131 
83032 
88941 
94857 

10-6500780 
06710 
12648 
18593 
24546 
30506 
36473 
42448 
48430 
54420 
60417 
66422 
72434 
78454 
84481 
90516 
96559 

10-6602609 
08667 
14733 
20806 
26887 
32976 
39073 
45177 
51289 
57409 

63537 
69673 
75817 
81969 
88128 
94296 
10-6700472 
06655 
12847 
19047 
25255 
120] 



dijf.] 

5767 

5774 
5781 
5788 
5795 
5801 
5809 
5816 
5823 
5829 
5837 
5844 
5851 
5859 
5865 
5872 



5887 
5894 
5901 
5909 
5916 
5923 
5930 
5938 
5945 
5953 
5960 
5967 
5975 
5982 
5990 
5997 
6005 
6012 
6020 
6027 
6035 
6043 
6050 

6058 
6066 
6073 
6081 
6089 
6097 
6104 
6112 
6120 
6128 
6136 
6144 
6152 
6159 
6168 
6176 
6183 
6192 
6200 
6208 

diff. 



■8° 

10-6725255 
31471 
37695 
43927 
50168 
56416 
62673 
68939 
75212 
81494 
87784 
94082 

10-6800389 
06705 
13028 
19360 
25701 
32050 



44774 

51149 
57532 
63924 
70325 
76734 
83152 
89579 
96015 

10-6902459 
08912 
15374 
21845 
28325 
34813 
41311 
47817 
54333 
60857 
67391 
73934 
80486 
87046 
93617 

10-7000196 
06784 
13382 
19989 
26605 
33231 
39866 
46511 
53164 
59828 
66500 
73183 
79874 
86576 
93287 

10-7100007 
06737 
13477 
110 

11^ 



diff. 

6216 
6224 
6232 
6241 
6248 
6257 
6266 
6273 
6282 
6290 

6298 
6307 
6316 
6323 
6332 
634] 
6349 
6358 
6366 
6375 
6383 
6392 
6401 
6409 
6418 
6427 
6436 
6444 
6453 
6462 
6471 
6480 
6488 
6498 
6506 
6516 
6524 
6534 
6543 
6552 
6560 
6571 
6579 
6588 
6598 
6607 
6616 
6626 
6635 
6645 
6653 
6664 
6672 
6683 
6691 
6702 
6711 
6720 
6730 
6740 

diff. 



10-7113477 
20227 
26986 
33755 
40534 
47323 
54122 
60930 
67749 
74577 

81415 
88264 
95122 

10-7201991 
08869 
15758 
22657 
29566 
36486 
43416 
50356 
57306 
64267 
71238 
78220 
85212 
92214 
99228 

10-7306251 
13286 
20331 
27387 
34453 
41530 
48618 
55717 
62827 
69947 
77079 
84221 

91375 
98539 
10-7405715 
12901 
20099 
27308 
34528 
41760 
49003 
56257 

63523 

70800 

78088 



92699 
10-7500022 
07357 
14703 
22061 
29431 
36812 
10° t 

LOG. COTAN. 



difi 

6750 
6759 
6769 
6779 
6789 
6799 
6808 
6819 
6828 
6838 

6849 
6858 
6869 
6878 
6889 
6899 
6909 
6920 
6930 
6940 
6950 
6961 
6971 
6982 
6992 
7002 
7014 
7023 
7035 
7045 
7056 
7066 
7077 
7088 
7099 
7110 
7120 
7132 

142 
7154 
7164 
7176 
7186 
7198 
7209 
7220 
7232 

243 
7254 
7266 
7277 
7288 
7300 
7311 
7323 
7335 
7346 
7358 
7370 
7381 



80° ^diff. 



10-7536812' 
442061 



7394 

7405 



5161174,7 
59028 '^^' 
66457 



73897 
81350 
88815 
96292 
10-7603782 

11283 
18797 
26322 
33861 
41411 
48974 
56549 
64137 
71738 
79350 
86976 
94614 
10-7702265 
09929 
17605 
25294 
32996 
40711 
48439 
56181 

63935 
71702 
79482 
87276 
95083 

10-7802903 
10736 
18583 
26444 
34317 
42205 
50106 
58020 
65949 
73891 
81847 
89816 
97800 

10-7905797 
13809 

21835 
29874 
37928 
45996 
54078 
62175 
70286 
78412 
86551 
94706 
10-8002875 
9° 



7429 
7440 
7453 
7465 
7477 
7490 
7501 

7514 
7525 
7539 
7550 
7563 
7575 
7588 
7601 
7612 
7626 
7638 
7651 
7664 
7676 
7689 
7702 
7715 
7728 
7742 
7754 
7767 
7780 
7794 
7807 
7820 
7833 
7847 
7861 
7873 
7888 
7901 
7914 
7929 
7942 
7956 
7969 
7964 
7997 
8012 
8026 

8039 
8054 
8068 
8082 
8097 
8111 
8126 
8139 
8155 
8169 



81° 
10-8002875 
11059 
19257 
27470 
35698 
43941 
52198 
60471 
68759 
77061 

85379 
93713 
10-8102061 
10425 
18804 
27198 
35608 
44034 
52475 
60932 

69405 
77894 



94918 
10-8203454 
12007 
20575 
29160 
37761 
46378 

55012 
63662 
72328 
81011 
89711 
98428 
10-8307161 
15911 
24678 
33462 

42263 
51081 
59917 
68769 
77639 
86527 
95431 

10-8404354 
13294 
22252 
31227 
40220 
49231 
58261 
67308 
76373 
85457 
94559 

10-8503679 
12818 
21975 
8° 



diff. 

8184 
8198 
8213 
8228 
8243 
8257 
8273 
8288 
8302 
8318 

8334 

8348 

8364 

8379 

8394 

8410 

8426 

8441 

845 

8473 

8489 
3504 
8520 
8536 
8553 
8568 
8585 
8601 
8617 
8634 
8650 
8666 
8683 
8700 
8717 
8733 
8750 
8767 
8784 



8818 
8836 
8852 
8870 
8888 
8904 
8923 
8940 
8958 
8975 
8993 
9011 
9030 
9047 
9065 
9084 
9102 
9120 
9139 
9157 

diff. 



126 



LOG. SINE. 



[Table n. ' 




1 
2 
3 

4 
5 

^ 

7 

8 

9 
10 
11 
12 
13 
14 
15 
16 
117 
,18 
19 
20 

;i 

122 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
38 
!39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



820 

9-9957 528 
705 
382 

9-9558 059 
235 
411 
586 
761 
936 

9-9%«lll 
284 
458 
631 
804 
977 

9-9960 149 
321 
492 
663 
834 

9-9961 004 
174 
343 
512 
681 
849 

9-9962 017 
185 
352 
519 
686 
852 

9-9963 018 
183 
348 
513 
677 
841 

9-9964 004 
167 
330 
493 
655 
816 
977 

9-9965 138 
299 
459 
619 
778 
937 

9-9966 096 
254 
412 
570 
727 
884 

9-9967 040 

196 

352 

507 

70 



177 
177 
177 
176 
176 
175 
175 
175 
175 
173 
174 
173 
173 
173 
172 
172 
171 
171 
171 
170 
170 
169 
169 
169 
168 
168 
168 
167 
167 
167 
166 
166 
165 
165 
165 
164 
164 
163 
163 
163 
163 
162 
161 
161 
161 
161 
160 
160 
159 
159 

159 
158 
158 
158 
157 
157 
156 
156 
156 
155 



83° 

9-9967 507 
662 
817 
971 

9-9968 125 
278 
431 
584 
736 
838 

9-9969 040 
191 
342 
492 
642 
792 
941 

9-9970 090 
239 
387 
535 
682 
829 
976 

9-9971 122 
268 
414 
559 
704 
849 
993 

9-9972 137 
280 
423 
566 
708 
850 
991 

9-9973 132 
273 
4W 
554 
693 
833 
971 

9-9974 110 
248 
386 
523 
660 
797 
933 

9-9975 069 
205 
340 
475 
609 
743 
877 

9-9976 Oil 
143 
6° 



diff. 

155 
155 
154 
154 
153 
153 
153 
152 
152 
152 
151 
151 
150 
150 
150 
149 
149 
149 
148 
148 
147 
147 
147 
146 
146 
146 
145 
145 
145 
144 
144 
143 
143 
143 
142 
142 
141 
141 
141 
141 
140 
139 
140 
138 
139 
138 
138 
137 
137 
137 
136 
136 
136 
135 
135 
134 
134 
134 
134 
132 



84° 

9-9976 143 
276 
408 
540 
672 
803 
933 

9-9977 064 
194 
323 
453 
582 
710 
838 
966 

9-9978 093 
220 
347 
473 
599 
725 
850 
975 

9-9979 099 
223 
347 
470 
593 
716 
838 
960 

9-9980 081 
202 
323 
44 
563 
683 
802 
921 

9-9981 040 
158 
275 
393 
510 
626 
743 
859 
974 

9-9982 

204 

318 
433 
546 
660 
772 
885 
997 
9-9983 109 
220 
332 
442 
50 



diff. 

133 
132 
132 
132 
131 
130 
131 
130 
129 
130 
129 
128 
128 
128 
127 
127 
1-27 
126 
126 
126 
125 
125 
124 
124 
124 
123 
123 
123 
122 
122 
121 
121 
121 
120 
120 
120 
119 
119 
119 
118 
117 
118 
117 
116 
117 
116 
115 
115 
115 
114 
115 
113 
114 
112 
113 
112 
112 
HI 
112 
110 



85° 

9-9983 442 
553 
663 
772 
881 
990 

9-9984 099 
207 
315 
422 
529 
636 
742 
848J 
953 

9-9985 058 
163 
268 
372 
475 
579 
682 
784 



090 
191 
292 
392 
492 
591 
691 
790 



diff 

111 
110 
109 
109 
109 
109 
108 
108 
107 
107 
107 
106 
106 
105 
105 
105 
105 
104 
103 
104 
103 
102 
102 
102 
102 
101 
101 
100 
100 

99 
100 

99 



9-9987 084 
181 
278 
375 
471 
567 
663 
758 
853 
947 

9-99S8 041 
135 
228 
321 
414 

506 



780 
871 
962 
9-9989 052 
141 
230 
319 
408 
40 



86° 

9-9989 408 
496 
584 
671 
758 
845 
931 

9-9990 017 
103 
188 
273 
357 
441 
525 
608 
691 
774 
856 
938 

9-9991 020 
101 
182 
282 
342 
422 
501 
580 
659 
737 
615 
892 
969 

9-9992 046 
122 
198 
274 
349 
424 
498 
572 
646 
720 
793 
865 
938 

9-9993 009 
081 
152 
223 
293 
364 
433 
503 
572 
640 
708 
776 
844 
911 
978 

9 9994 044 
30 



88 
83 
87 
87 
87 
86 
86 
86 
85 
85 
84 
84 
84 
33 
83 
83 
82 
82 
82 
81 
81 
SO 
80 
80 
79 
79 
79 
78 
78 
77 

77 
77 
76 
76 
76 
75 
75 
74 
74 
74 
74 
73 
72 
73 
71 
72 
71 
71 
70 
71 
69 
70 
69 
68 
68 
68 
68 
67 
67 
66 



60; 

59i 
58! 
57: 
56| 
55! 
54l 
53: 
521 
51j 

50' 

4P| 
48! 
47: 

46; 

451 
44i 
43 
42 
41 

401 

39| 
38 
37 
36 
35i 
34: 
331 
32| 
3l| 

30l 
29 
28 
27 

:6 

25 
24 
23 

22 

21, 

20 
19 
18| 
17 
16 
15 
14 
13 
12 
11 
10 



LOG. COSINE. 



Table ii.] 



LOG. TAN. 



127 



82° 

10-65 21975 
31151 
40345 
49558 
58790 
68041 
77311 
86600 
95908 

10-66 05236 
14583 
23949 
33335 
42740 
52165 
61609 
71074 
80558 
90063 
99587 

10-87 09132 
1869 
28282 
37888 
47514 
57161 
66829 
76518 
86227 
95957 

10-88 05709 
15482 
25276 
35091 
44928 
54787 
64667 
74569 
84492 
94438 

10-8904406 
14396 
24409 
34443 
44500 
54580 
64683 
74808 
84956 
95128 

10-9005322 
15540 
25781 
36045 
46333 
56645 
66980 
77340 
87723 
98131 

10-9i 08562 
70 



9176 
9194 
9213 
9232 
9251 
9270 
9289 
9308 
9328 
9347 

9366 
9386 
9405 
9425 
9444 
9465 
■9484 
9505 
9524 
9545 

9565 
9585 
9606 
9626 
9647 
9668 
9689 
9709 
9730 
9752 

9773 
9794 
9815 
9837 
9859 
9880 
9902 
9923 
9946 
9968 
9990 
10013 
10034 
10057 
10080 
10103 
10125 
10H8 
10172 
10194 

10218 
10241 
10264 
10288 
10312 
10335 
10360 
10383 
10408 
10431 



83° 

10-91 08562 
19019 
29499 
40004 
50534 
61089 
71669 
82274 
92904 

10-92 03559 
14240 
24947 
35679 
46437 
57221 
68031 
78867 
89730 

10-93 00619 
11535 

22478 
33447 
44444 
55467 
66518 
77597 
88703 
99836 
10-94 10998 
22187 

33405 
44651 
55926 
67229 
78561 
89922 
10-9501311 
12730 
24179 
35657 

47164 
58701 
70269 
81866 
93494 
10-96 05152 
16841 
28561 
40312 
52094 

63907 
75751 
87627 
99536 
10-97 11476 
23448 
35452 
47490 
59559 
71662 
83798 
6° 



di 

10457 
10480 
10505 
10530 
10555 
10580 
10605 
10630 
10655 
10681 
10707 
10732 
10758 
10784 
10810 
10836 
10863 
10889 
10916 
10943 

10969 
10997 
11023 
11051 
11079 
11106 
11133 
11162 
11189 
11218 

11246 
11275 
11303 
11332 
11361 
11389 
11419 
11449 
11478 
11507 
11537 
11568 
11597 
11628 
11658 
11689 
11720 
11751 
11782 
11813 
11844 
11876 
11909 
11940 
11972 
12004 
12038 
12069 
12103 
12136 

dijf. 



84° 

10-97 83798 
95967 

10-98 08169 
20406 
32675 
44979 
57318 
69690 
82097 
94539 

10-99 07016 
19529 
32076 
44660 
57279 
69934 
826-25 
95353 

11-00 08117 
20919 
33757 
46633 
59546 
72497 
85486 
98513 

11-01 11579 
24683 
37827 
51009 
64231 
77493 
.90794 

11-02 04135 
17517 
30940 
44403 
57908 
71453 
85041 

98670 
11-03 12342 
26056 
39812 
53612 
67455 
81341 
95272 
11-04 09246 
23265 

37328 
51436 
65590 
79789 
94033 
11-05 08324 
22662 
37046 
51477 
65956 
80482 
5° 



diff. 

12169 
12202 
12237 
12269 
12304 
12339 
12372 
12407 
12442 
12477 

12513 
12547 
12584 
12619 
12655 
12691 
12728 
12764 
12802 
12838 
12876 
12913 
12951 
12989 
13027 
13066 
13104 
13144 
13182 
13222 

13262 
13301 
13341 

13382 
13423 
13463 
13505 
13545 
13588 
13629 
13672 
13714 
13756 
13800 
13843 
13886 
13931 
13974 
14019 
14063 
14108 
14154 
14199 
14244 
14291 
14338 
14384 
14431 
14479 
14526 



85° 

•05 80482 
95056 

■06 09679 
24350 
39071 
53840 
68660 
83529 
98448 

•07 13419 
28440 
43513 
58637 
73814 
89043 

•08 04325 
19660 
35048 
50491 
65988 
81540 
97147 

•09 12810 
28528 
44303 
60134 
76023 
91970 

-10 07974 
24037 

40158 
56340 

72580 



•11 05243 
21666 
38150 
54697 
71306 
87978 

•12 04714 
21513 
38377 
55306 
72301 
89362 

•13 06489 
23683 
40945 
58275 

75673 
93141 

•14 10679 
28287 
45966 
63717 
81539 
99434 

•15 17403 
35446 
53563 
40 



diff. 

Uhl^ 
14623 
14671 
14721 
14769 
14820 
14869 
14919 
14971 
15021 

15073 
15124 
15177 
15229 
15282 
15335 
15368 
15443 
15497 
15552 

15607 
15663 
15718 
15775 
15831 
15889 
15947 
16004 
16063 
16121 
16182 
16240 
16301 
16362 
16423 
16484 
16547 
16609 
16672 
16736 

16799 
16864 
16929 
16995 
17061 
17127 
17194 
17262 
17330 
17398 
17468 
17538 
17608 
17679 
17751 
17822 
17895 
17969 
18043 
18117 

diff 



LOG. COTAN. 



128 LOG. SINE. 



LOG. TAN. [Table ii. 



/ 


87° !< 





9-9994 044 


1 


110 


2 


176 


3 


241 


4 


306 


5 


370 


6 


435 


7 


498 


8 


562 


9 


625 


10 


688 


11 


750 


12 


812 


13 


874 


14 


935 


15 


996 


16 


9-9995 056 


17 


116 


18 


176 


19 


236 


20 


295 


21 


353 


22 


411 


23 


469 


24 


527 


25 


584 


26 


641 


27 


697 


28 


753 


29 


809 


30 


865 


31 


919 


32 


974 


33 


9-9995 028 


34 


082 


35 


136 


36 


189 


37 


242 


38 


294 


39 


346 


40 


398 


41 


449 


42 


500 


43 


550 


44 


601 


45 


650 


46 


700 


47 


749 


48 


798 


49 


846 


50 


894 


51 


942 


52 


989 


53 


9.9997 036 


54 


082 


55 


128 


56 


174 


57 


220 


58 


265 


59 


309 


f)0 


354 


' 


2o 



LOG. COSINE. 



^ 



"7^ 



10 

11 

12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 

149 
50 
.51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



86° 
11-15 53563 

71755 
90023 

11-16 08367 
26789 
45288 
63866 
82522 

11-17 01259 
20076 
38974 
57954 
77016 
96162 

11-18 15392 
34706 
54106 
73593 
93166 

11-19 12828 
32578 
52417 
72347 
92368 

11-20 12481 
32687 
52985 
73330 
93870 

11-21 14456 

35139 
55921 
75801 
9778: 

11-22 18854 
40048 
61335 
82726 

11-23 04223 
25825 

47535 
69353 
91281 

11-24 13319 
35469 
57731 
80108 

11-25 02600 
25208 
47933 

70778 
93742 

11-26 16828 
40036 
63369 
86826 

11-27 10411 
34123 
57965 
81937 

11-28 05042 
3° 



diff. 

18192 
18268 
18344 
18422 
18499 
18578 
18556 
18737 
18817 
18898 

18980 

19052 

19146 

19230 

19314 

19400 

1948 

19573 

19652 

19750 

19839 
19930 
20021 
20113 
20206 
20299 
20394 
20490 
20586 
20683 
20782 



20981 
21082 
21184 
21287 
21391 
21497 
21602 
21710 
21818 
21928 
22038 
22150 
22252 
22377 
22492 
22608 
22725 
22845 
22964 
23085 
23208 
23333 
23457 
23585 
23712 
23842 
23972 
24105 

diff] 



87° ' 

11-28 06042 
30281 
54655 
79156 

11-29 03815 
28605 
53535 
78510 

11-30 03828 
29194 
54708 
80371 

11-3106187 
32156 
58281 
84553 

11-32 11004 
37607 
64372 
91303 

11-33 18402 
45669 
73109 

11-34 00721 
28510 
55478 
84625 

11-35 12955 
41472 
70175 

99059 

11-36 28155 
57437 
85917 

11-37 15598 
45482 
76573 

11-38 06873 
37384 
68111 
99057 

11-39 30223 
61614 
93233 

11-40 25083 
57158 
89491 

11-41 22055 
54864 
87923 

11-42 21234 
54803 
88532 

11-43 22725 
57088 
91724 

11-44 26638 
61834 
97317 

11-45 33091 
69162 
2° 



diff. 

24239 
24374 
24511 
24649 
24790 
24930 
25075 
25218 
25366 
25514 

25653 
25816 
25969 
26125 
26282 
26441 
26603 
25765 
26931 
27099 
27267 
27440 
27612 
27789 
27968 
28147 
28331 
28515 
28703 
28894 
29086 
29282 
29480 
29681 
29884 
30091 
30300 
30511 
30727 
30945 
31166 
31391 
31619 
31850 
32085 
32323 
32564 
32809 
33059 
33311 

33569 
33829 
34093 
34363 
34636 
34914 
35196 
35483 
35774 
35071 



LOG. COTAN. 



130 LOG. SINE 88°. [Table iuj 




// 


C 


V 


2' 


3' 


4' 


5' 


6' 


r >"ll 







5-9997354 


9997398 


9997441 


9997484 


•9997527 


•9997570 


•9997612 


9-9997653 


60 






54 


^ 98 


^ 42 


^ 85 


^ 28 


a. 70 


<^ 12 


54 


59 




2 


55 


99 


43 


86 


29 


71 


13 


55 


58 




3 


56 


9997400 


43 


86 


29 


72 


14 


55 


57 




4 


57 


00 


44 


87 


30 


72 


14 


56 


56 




5 


57 


01 


45 


88 


31 


73 


15 


57 


55 




6 


58 


02 


45 


89 


31 


74 


16 


58 


54 




7 


59 


03 


46 


89 


32 


74 


17 


58 


53 




8 


59 


03 


47 


90 


33 


75 


17 


59 


52 




9 


9-9997360 


9997404 


9997448 


9997491 


•9997534 


•9997576 


-9997618 


9^9997660 


51 




10 


61 


05 


48 


91 


34 


77 


19 


60 


50 




11 


62 


06 


49 


92 


35 


77 


19 


61 


49 




12 


62 


06 


49 


93 


36 


78 


20 


62 


48 




13 


63 


07 


50 


94 


36 


79 


21 


62 


47 




14 


64 


08 


51 


94 


37 


79 


21 


63 


46 




15 


65 


08 


52 


95 


38 


80 


22 


64 


45 




16 


65 


09 


53 


96 


38 


81 


23 


64 


44 




17 


66 


10 


53 


96 


39 


82 


24 


65 


43 




18 


67 


11 


54 


97 


40 


82 


24 


66 


42 




19 


3-9997368 


9997411 


•9997455 


•9997498 


-9997541 


•9997583 


-9997625 


9-9997667 


41 




20 


68 


12 


56 


99 


41 


84 


26 


67 


40 




21 


69 


13 


56 


99 


42 


84 


26 


68 


39 




22 


70 


14 


57 


•9997500 


43 


85 


27 


69 


38 




23 


70 


14 


58 


01 


43 


86 


28 


69 


37 




24 


71 


15 


58 


01 


44 


86 


28 


70 


36 




25 


72 


16 


59 


02 


45 


87 


29 


71 


35 




26 


73 


16 


60 


03 


46 


88 


30 


71 


34 




27 


73 


17 


61 


04 


46 


89 


30 


72 


33 




28 


74 


18 


61 


04 


47 


89 


31 


73 


32 




29 


9-9997375 


9997419 


•9997462 


•9997505 


-9997548 


•9997590 


-9997632 


9-9997673 


31 




30 


76 


19 


63 


06 


48 


91 


33 


74 


30 




31 


76 


20 


63 


06 


49 


91 


33 


75 


29 




32 


77 


21 


64 


07 


50 


92 


34 


75 


28 




33 


78 


22 


65 


08 


51 


93 


35 


76 


27 




34 


79 


22 


66 


09 


51 


93 


35 


77 


26 




35 


79 


23 


66 


09 


52 


94 


36 


78 


25 




36 


80 


24 


67 


10 


53 


95 


37 


78 


24 




37 


81 


24 


68 


11 


53 


96 


37 


79 


23 




38 


81 


25 


68 


11 


54 


96 


38 


80 


22 




39 


9-9997382 


•9997426 


•9997469 


-9997512 


•9997555 


•9997597 


•9997639 


9-9997680 


21 




40 


83 


27 


70 


13 


55 


98 


40 


81 


20 




41 


84 


27 


71 


14 


56 


98 


40 


82 


19 




42 


84 


28 


71 


14 


57 


99 


41 


82 


18 




43 


85 


29 


72 


15 


58 


-9997600 


42 


83 


17 




44 


86 


30 


73 


16 


58 


00 


42 


84 


16 




45 


87 


30 


74 


16 


59 


01 


43 


84 


15 




46 


87 


31 


74 


17 


60 


02 


44 


85 


14 




47 


88 


32 


75 


18 


60 


03 


44 


86 


13 




48 


89 


32 


76 


19 


61 


03 


45 


87 


12 




49 


9-9997389 


-9997433 


•9997476 


•9997519 


•9997562 


•9997604 


-9997646 


9^9997687 


11 




1 50 


90 


34 


77 


20 


63 


05 


46 


88 


10 




51 


91 


35 


78 


21 


63 


05 


47 


89 


9 




52 


92 


35 


79 


21 


64 


06 


49 


89 


8 




53 


92 


36 


79 


22 


65 


07 


49 


90 


7 




54 


93 


37 


80 


23 


65 


07 


49 


91 


6 




55 


94 


37 


81 


24 


66 


08 


50 


91 


5 




56 


95 


38 


81 


24 


67 


09 


51 


92 


4 




57 


95 


39 


82 


25 


67 


IC 


51 


93 


3 




58 


96 


4C 


83 


2€ 


6e 


IC 


52 


93 


2 




59 


9-9997397 


-999744C 


•9997484 


•999752f 


) -9997569 


-9997611 


-999765S 


9-9997694 


1 




60 


9£ 


41 


84 


27 7C 


i:; 


5S 


95 







59' 


58' 


57' 


56' 55' 


54' 


53' 


52' 


// 




LOG. COSINE 1°. 





Table II.] 


LOG. TAN. 88°. 


131 




// 


0' . 1' 


2' 


3' 4' 


5' 


6' 


7' 


" 







11-4569162 


4605534 


4642213 


4679203 4716510 


4754140 


4792098 


11-4830390 


60 




1 


9766 


^ 6143 


^ 2827 


9822 


^ 7135 


^ 4770 


:-. 2733 


1031 


59 




2 


11-4570369 


H 6752 


::! 3441 


4680441 


Ci 7759 


CL 5400 


^ 3369 


1672 


58 




3 


0973 


7361 


4055 


^ 1061 


8384 


6030 


4005 


2313 


57 




4 


1577 


7970 


4669 


ZL 1680 


9009 


6660 


4640 


2955 


56 




5 


2181 


8579 


5283 


2300 


9634 


7291 


5276 


3596 


55 




6 


2786 


9188 


5898 


2919 


4720259 


7921 


5912 


4238 


54 




7 


3390 


9797 


6512 


3539 


0884 


8551 


6548 


4879 


53 




8 


3994 


4610407 


7127 


4159 


1509 


9182 


7184 


5521 


52 




9 


4598 


1016 


7741 


4779 


2134 


9813 


7820 


6163 


51 




10 


5203 


1626 


8356 


5399 


2759 


4760443 


8457 


6805 


50 




11 


5807 


2235 


8971 


6019 


3385 


1074 


9093 


7447 


49 




12 


6412 


2845 


9586 


6639 


4010 


1705 


9729 


8089 


48 




13 


7017 


3455 


4650201 


7259 


4636 


2336 


4800366 


8731 


47 




14 


7622 


4065 


0816 


7879 


5261 


2967 


1003 


9373 


46 




15 


8227 


4675 


1431 


8500 


5887 


3599 


1639 


11-4840016 


45 




16 


8832 


5285 


2046 


9120 


6513 


4230 


2276 


0658 


44 




17 


9437 


5895 


2661 


9741 


7139 


4861 


2913 


1301 


43 




18 


11-4580042 


6505 


3277 


4690362 


7765 


5493 


3550 


1943 


42 




19 


0647 


7116 


3892 


0982 


8391 


6124 


4187 


2586 


41 




20 


1252 


7726 


4508 


1603 


9017 


6756 


4825 


3229 


40 




21 


1858 


8336 


5124 


2224 


9644 


7388 


5462 


3872 


39 




22 


2463 


8947 


5739 


2845 


4730270 


8020 


6099 


4515 


38 




23 


3069 


9558 


6355 


3466 


0897 


8651 


6737 


5158 


37 




24 


3674 


4620168 


6971 


4088 


1523 


9283 


7374 


5801 


36 




25 


4280 


0779 


7587 


4709 


2150 


9916 


8012 


6445 


35 




26 


4886 


1390 


8203 


5330 


2777 


4770548 


8650 


7088 


34 




27 


5492 


2001 


8819 


5952 


3403 


1180 


9288 


7732 


33 




28 


6098 


2612 


9436 


6573 


4030 


1813 


9926 


8375 


32 




29 


6704 


3223 


4660052 


7195 


4657 


2445 


4810564 


9019 


31 




30 


7310 


3835 


0669 


7817 


5284 


3078 


1202 


9663 


30 




31 


7916 


4446 


1285 


8438 


5912 


3710 


1840 


11-4850307 


29 




32 


8523 


5058 


1902 


9060 


6539 


4343 


2478 


0951 


28 




33 


9129 


5669 


2518 


9682 


7166 


4976 


3117 


1595 


27 




34 


9736 


6281 


3135 


4700304 


7794 


5609 


3755 


2239 


26 




35 


11-4590342 


6892 


3752 


0927 


8421 


6242 


4394 


2883 


25 




36 


0949 


7504 


4369 


1549 


9049 


6875 


5033 


3528 


24 




37 


1555 


8116 


4986 


2171 


9677 


7508 


5671 


4172 


23 




38 


2162 


8728 


5603 


2794 


4740305 


8142 


6310 


4817 


22 




39 


2769 


9340 


6621 


3416 


0933 


8775 


6949 


5461 


21 




40 


3376 


9952 


6838 


4039 


1561 


9409 


7588 


6106 


20 




41 


3983 


4630564 


7455 


4662 


2189 


4780042 


8228 


6751 


19 




42 


4591 


1177 


8073 


5284 


2817 


0676 


8867 


7396 


18 




43 


5198 


1789 


8690 


5907 


3445 


1310 


9506 


8041 


17 




44 


5805 


2401 


9308 


6530 


4074 


1943 


4820146 


8686 


16 




45 


6413 


3014 


9926 


7153 


4702 


2577 


0785 


9332 


15 




46 


7020 


3627 


4670544 


7777 


5331 


3211 


1425 


9977 


14 




47 


7628 


4239 


1162 


8400 


5959 


3846 


2065 


11-4860622 


13 




48 


8235 


4852 


1780 


9023 


6588 


4480 


2704 


1268 


12 




49 


8843 


5465 


2398 


9647 


7217 


5114 


3344 


1913 


11 




50 


9451 


6078 


3016 


4710270 


7846 


5749 


3984 


2559 


10 




51 


11-4600059 


6691 


3634 


0894 


8475 


6383 


4625 


3205 


9 




52 


0667 


7304 


4253 


1517 


9104 


7018 


5265 


3851 


8 




53 


1275 


7918 


4871 


2141 


9733 


7652 


5905 


4497 


7 




54 


1883 


8531 


5490 


2765 


4750362 


8287 


6545 


5143 


6 




55 


2491 


9144 


6108 


3389 


0992 


8922 


7186 


5789 


5 




56 


3100 


9758 


6727 


4013 


1621 


9557 


7827 


6436 


4 




57 


3708 


4640371 


7346 


4637 


2251 


4790192 


8467 


7082 


3 




58 


4317 


0985 


7965 


5261 


2880 


0827 


9108 


7728 


2 




59 


4925 


1599 


8584 


5886 


3510 


1463 


9749 


8375 


1 




60 


5534 


2213 


9203 


6510 4140 


2098 


4830390 


9022 






59' 


58' 


57' 


56' 55' 


54' 


53' 


52' 




L 




LOG. COTAN. 1°. 







132 LOG. SINE 88°. [Table n. 11 


// 


8' 


9' 


10' 


11' 


12' 


13' 


14' 


15' I 


// 





9-9997695 


9997736 


•9997776 


•9997817 


•9997856 


•9997896 


-9997935 


9^9997974 


60 


1 


95 


^ 36 


a> 77 


a. 17 


=^ 57 


a. 97 


^ 36 


75 


59 


2 


96 


37 


78 


18 


58 


97 


36 


75 


53 


3 


97 


38 


78 


19 


58 


98 


37 


76 


57 


4 


97 


38 


79 


19 


59 


99 


38 


77 


56 


5 


98 


39 


80 


20 


60 


99 


38 


77 


55 


6 


99 


40 


80 


21 


60 


•9997900 


39 


78 


54 


7 


9-9997700 


40 


81 


21 


61 


01 


40 


78 


53 


8 


00 


41 


82 


22 


62 


01 


40 


79 


52 


9 


9-9997701 


9997742 


•9997782 


9997823 


•9997862 


•9997902 


•9997941 


9^9997980 


51 


10 


02 


43 


83 


23 


63 


03 


42 


80 


50 


11 


02 


43 


84 


24 


64 


03 


42 


81 


49 


12 


03 


44 


84 


25 


64 


04 


43 


82 


48 


13 


04 


45 


85 


25 


65 


05 


44 


82 


47 


14 


04 


45 


86 


26 


66 


05 


44 


83 


46 


15 


05 


46 


86 


27 


66 


06 


45 


84 


45 


16 


06 


47 


87 


27 


67 


06 


46 


84 


44 


J7 


06 


47 


88 


28 


68 


07 


46 


85 


43 


18 


07 


48 


88 


29 


68 


08 


47 


86 


42 


19 


9-9997708 


9997749 


•9997789 


•9997829 


•9997869 


•9997908 


•9997947 


9^9997986 


41 


20 


08 


49 


90 


30 


70 


09 


48 


87 


40 


21 


09 


50 


90 


31 


70 


10 


49 


87 


39 


22 


10 


51 


91 


31 


71 


10 


49 


88 


38 


23 


10 


51 


92 


32 


72 


11 


50 


89 


37 


24 


11 


52 


93 


33 


72 


12 


51 


89 


36 


25 


12 


53 


93 


33 


73 


12 


51 


90 


35 


26 


13 


53 


94 


34 


74 


13 


52 


91 


34 


27 


13 


54 


95 


35 


74 


14 


53 


91 


33 


28 


14 


55 


95 


35 


75 


14 


53 


92 


32 


29 


9-9997715 


9997755 


•9997796 


•9997836 


•9997876 


•9997915 


•9997954 


9^9997993 


31 


30 


15 


56 


97 


37 


76 


16 


55 


93 


30 


31 


16 


57 


97 


37 


77 


16 


55 


94 


29 


32 


17 


57 


98 


38 


78 


17 


56 


94 


28 


33 


17 


58 


99 


39 


78 


18 
18 


57 


95 


27 


34 


18 


59 


99 


39 


79 


57 


96 


26 


35 


19 


59 


•9997800 


40 


80 


19 


58 


96 


25 


36 


19 


60 


01 


41 


80 


20 


58 


97 


24 


37 


20 


61 


01 


41 


81 


20 


59 


98 


23 


38 


21 


62 


02 


42 


82 


21 


60 


98 


22 


39 


9-9997721 


•9997762 


•9997803 


-9997843 


•9997882 


•9997921 


•9997960 


9^9997999 


21 


40 


22 


63 


03 


43 


83 


22 


61 


9^9998000 


20 


41 


23 


64 


04 


44 


84 


23 


62 


00 


19 


42 


23 


64 


05 


45 


84 


23 


621 01 


18 


43 


24 


65 


05 


45 


85 


24 


63 


02 


17 


44 


25 


66 


06 


46 


86 


25 


64 


02 


16 


45 


26 


66 


07 


47 


86 


25 


64 


03 


15 


46 


26 


67 


07 


47 


87 


26 


65 


03 


14 


47 


27 


68 


08 


48 


87 


27 


66 


04 


13 


48 


28 


68 


m 


49 


88 


27 


66 


05 


12 


49 


9-9997728 


•9997769 


•9997809 


•9997849 


•9997839 


•9997928 


•9997967 


9-9998005 


11 


50 


29 


70 


10 


50 


89 


29 


68 


06 


10 


51 


30 


70 


11 


51 


90 


29 


68 


07 


9 


52 


30 


71 


11 


51 


91 


30 


69 


07 


8 


53 


31 


72 


12 


52 


91 


31 


70 


08 


7 


54 


32 


72 


13 


53 


92 


31 


70 


09 


6 


55 


32 


73 


13 


53 


93 


32 


71 


09 


5 


56 


33 


74 


14 


54 


93 


33 


71 


10 


4 


57 


34 


74 


15 


55 


94 


33 


72 


10 


3 


58 


34 


75 


15 


55 


95 


34 


73 


11 


2 


59 


9-9997735 


•9997776 


•9997816 


■9997856 


•9997895 


•9997935 


•9997973 


9^9998012 


1 


60 


36 


76 


17 


56 


96 


35 


74 


12 





" 


51' 


50' 


49' 


48' 


47' 


46' 


1 45' 


44' 


" 


LOG. COSINE 1°. j| 



i Table ii.] log. tan. 88°. 




133 1 


// 


8^ 


9' 


10' 


\V 


12' 


13' 


14' 


15' 


" 





11-4869022 


4907999 


4947329 


4987018 


5027072 


5067498 


5108304 


11-5149495 


60 


1 


9668 


^ 8652 


^ 7988 


^ 7683 


^ 7743 


U 8175 


^ 8987 


11-5150185 


59 


2 


11-4870315 


^ 9305 


^ 8647 


^ 8347 


^ 8414 


^ 8852 


"^ 9670 


0875 


58 


3 


0962 


9958 


9305 


9012 


9085 


9529 


5110354 


1565 


57 


4 


1609 


4910610 


9964 


9677 


9756 


5070207 


1038 


2255 


56 


5 


2257 


1263 


4950623 


4990342 


5030427 


0884 


1721 


2945 


55 


6 


2904 


1916 


1282 


1007 


1098 


1562 


2405 


3636 


54 


7 


3551 


2570 


1941 


1672 


1769 


2239 


3089 


4326 


53 


8 


4199 


3223 


2600 


2337 


2441 


2917 


3773 


5017 


52 


9 


4846 


3876 


3260 


3003 


3112 


3595 


4457 


5708 


51 


10 


5494 


4530 


3919 


3668 


3784 


4273 


5142 


6398 


50 


11 


6141 


5183 


4579 


4334 


4456 


4951 


5826 


7089 


49 


12 


6789 


5837 


5238 


5000 


5127 


5629 


6511 


7780 


48 


13 


7437 


6491 


5898 


5665 


5799 


6307 


7195 


8472 


47 1 


14 


8085 


7145 


6558 


6331 


6471 


6985 


7880 


9163 


46 


15 


8733 


7799 


7218 


6997 


7144 


7664 


8565 


9854 


45 


16 


9382 


8453 


7878 


7663 


7816 


8342 


9250 


11-5160546 


44 


17 


11-4880030 


9107 


8538 


8329 


8488 


9021 


9935 


1237 


43 


18 


0678 


9761 


9198 


8996 


9161 


9700 


5120620 


1929 


42 


19 


1327 


4920416 


9858 


9662 


9833 


5080379 


1305 


2621 


41 


20 


1975 


1070 


4960519 


5000329 


5040506 


1058 


1991 


3313 


40 


21 


2624 


1725 


1179 


0995 


1179 


1737 


2676 


4005 


39 


22 


3273 


2379 


1840 


1662 


1852 


2416 


3362 


4697 


38 


23 


3922 


3034 


2501 


2329 


2524 


3095 


4048 


5389 


37 


24 


4571 


3689 


3162 


2996 


3198 


3774 


4733 


6081 


36 


25 


5220 


4344 


3822 


3663 


3871 


4454 


5419 


6774 


35 


26 


5869 


4999 


4483 


4330 


4544 


5134 


6105 


7467 


34 


27 


6518 


5654 


5145 


4997 


5217 


5813 


6791 


8159 


33 


28 


7168 


6309 


5806 


5664 


5891 


6493 


7478 


8852 


32 


29 


7817 


6965 


6467 


6332 


6565 


7173 


8164 


9545 


31 


30 


8467 


7620 


7129 


6999 


7238 


7853 


8851 


11-5170238 


30 


31 


9117 


8276 


7790 


7667 


7912 


8533 


9537 


0931 


29 


32 


9766 


8931 


8452 


8334 


8586 


9213 


5130224 


1624 


28 


33 


11-4890416 


9587 


9113 


9002 


9260 


9894 


0911 


2318 


27 


34 


1066 


0243 


9775 


9670 


9934 


5090574 


1597 


3011 


26 


35 


1716 


4930899 


4970437 


501033S 


5050608 


1255 


2284 


3705 


25 


36 


2366 


1555 


1099 


1006 


1283 


1935 


2972 


4398 


24 


37 


3017 


2211 


1761 


1675 


1957 


2616 


3659 


5092 


23 


38 


3667 


2867 


2424 


2343 


2632 


3297 


4346 


5786 


22 


39 


4317 


3523 


3086 


3011 


3306 


3978 


5034 


6480 


21 


40 


4968 


4180 


3748 


3680 


3981 


4659 


5721 


7174 


20 


41 


5619 


4836 


4411 


4348 


4656 


5340 


6409 


7869 


19 


42 


6269 


5493 


5073 


5017 


5331 


6022 


7097 


8563 


18 


43 


6920 


6150 


5736 


5686 


6006 


6703 


7784 


9257 


17 


44 


7571 


6807 


6399 


6355 


6681 


7385 


8472 


9952 


16 


45 


8222 


7464 


7062 


7024 


7357 


8066 


9161 


11-5180647 


15 


46 


8873 


8121 


7725 


7693 


8032 


8748 


9849 


1341 


14 


47 


9525 


8778 


8388 


8362 


8707 


9430 


5140537 


2036 


13 


48 


11-4900176 


9435 


9051 


9032 


9383 


5100112 


1225 


2731 


12 


49 


0827 


4940092 


9715 


9701 


5060059 


0794 


1914 


3426 


11 


50 


1479 


0750 


4980378 


5020371 


0734 


1476 


2603 


4122 


10 


51 


2130 


1407 


1042 


1041 


1410 


2158 


3291 


4817 


9 


52 


2782 


2065 


1705 


1710 


2086 


2841 


3980 


5513 


8 


53 


3434 


2723 


2369 


23^0 


2763 


3523 


4669 


6208 


7 


54 


4086 


3380 


3033 


3050 


3439 


4206 


5358 


6904 


6 


55 


4738 


4038 


3697 


3720 


4115 


4888 


6047 


7600 


5 


56 


5390 


4696 


4361 


4390 


4792 


5571 


6737 


8296 


4 


57 


6042 


5354 


5025 


5061 


5468 


6254 


7426 


8992 


3 


58 


6695 


6013 


5689 


5731 


6145 


6937 


8116 


96SS 


2 


59 


7347 


6671 


6354 


6402 


6821 


7620 


8805 


11-5190384 


1 


60 


7999 


7329 


7018 


7072 


7498 


8304 


9495 


1080 





" 


5r 


50' 


49' 


48' 


47' 


46' 


45' 


44' 


" 





LOG. COTAN. \o. 




Ii 



12 



134 LOG. SINE 88°. [TaWeii. II 


// 


16' 


IT 


18' 


19' 


20' 


21' 


22' 


23' 


// 





9-9998012 


9998050 


•9998088 


•9998125 


•9998162 


9998199 


•9998235 


9^9998271 


60 


1 


13 


EL 51 


El 89 


EL 26 


EL 63 


EL 99 


^ 36 


72 


59 


2 


14 


52 


89 


27 


64 


9998200 


36 


72 


58 


3 


14 


52 


90 


27 


64 


01 


37 


73 


57 


4 


15 


53 


91 


28 


65 


01 


38 


73 


56 


5 


16 


54 


91 


28 


65 


02 


38 


74 


55 


6 


16 


54 


92 


29 


66 


03 


39 


75 


54 


7 


17 


55 


92 


30 


67 


03 


39 


75 


53 


8 


17 


55 


93 


30 


67 


04 


40 


76 


52 


9 


9-999S018 


9998056 


-9998094 


•9998131 


-9998168 


-9998204 


•9998241 


9.9998276 


51 


10 


19 


57 


94 


32 


68 


05 


41 


77 


50 


11 


19 


57 


95 


32 


69 


06 


42 


77 


49 


12 


20 


58 


96 


33 


70 


06 


42 


78 


48 


13 


21 


59 


96 


33 


70 


07 


43 


79 


47 


14 


21 


59 


97 


34 


71 


07 


44 


79 


46 


15 


22 


60 


97 


35 


72 


08 


44 


80 


45 


16 


23 


60 


98 


35 


72 


09 


45 


60 


44 


17 


23 


61 


99 


36 


73 


09 


45 


81 


43 


18 


24 


62 


99 


37 


73 


10 


46 


82 


42 


19 


9-9998024 


9998062 


•9998100 


•9998137 


•9998174 


-9998210 


•9998-246 


9^9998282 


41 


20 


25 


63 


01 


38 


75 


11 


47 


83 


40 


21 


26 


64 


01 


38 


75 


12 


48 


83 


39 


22 


26 


64 


02 


39 


76 


12 


48 


84 


38 


23 


27 


65 


02 


40 


76 


13 


49 


85 


37 


24 


28 


66 


03 


40 


77 


13 


49 


85 


36 


25 


28 


66 


04 


41 


78 


14 


50 


86 


35 


26 


29 


67 


04 


41 


78 


15 


51 


86 


34 


27 


30 


67 


05 


42 


79 


15 


51 


87 


33 


28 


30 


68 


06 


43 


79 


16 


52 


88 


32 


29 


9-9998031 


9998069 


-9998106 


•9998143 


•9998180 


-9998216 


•9998252 


9-9998288 


31 


30 


31 


69 


07 


44 


81 


17 


53 


89 


30 


31 


32 


70 


07 


45 


81 


18 


54 


89 


29 


32 


33 


71 


08 


45 


82 


18 


54 


90 


28 


33 


33 


71 


09 


46 


82 


19 


55 


91 


27 


34 


34 


72 


09 


46 


83 


19 


55 


91 


26 


35 


35 


72 


10 


47 


84 


20 


56 


92 


25 


36 


35 


73 


11 


48 


84 


21 


57 


92 


24 


37 


36 


74 


11 


48 


85 


21 


57 


93 


23 


38 


36 


74 


12 


49 


86 


22 


58 


93 


22 


39 


9-9998037 


•9998075 


•9998112 


•9998149 


•9998186 


•9998222 


•9998258 


9-9998294 


21 


40 


38 


76 


13 


50 


87 


23 


59 


95 


20 


41 


38 


76 


14 


51 


87 


24 


60 


95 


19 


42 


39 


77 


14 


51 


88 


24 


60 


96 


18 


43 


40 


77 


15 


52 


89 


25 


61 


96 


17 


44 


40 


78 


15 


53 


89 


25 


61 


97 


16 


45 


41 


79 


16 


53 


90 


26 


62 


98 


15 


46 


42 


79 


17 


54 


90 


27 


63 


98 


14 


47 


42 


80 


17 


54 


91 


27 


63 


99 


13 


48 


43 


81 


18 


55 


92 


28 


64 


9-9998300 


12 


49 


9-9998043 


•9998081 


•9998119 


•9998156 


-9998192 


•9998229 


•9998264 


9-9998300 


11 


50 


44 


82 


19 


56 


93 


29 


65 


01 


10 


51 


45 


82 


20 


57 


93 


30 


66 


01 


9 


52 


45 


83 


20 


57 


94 


30 


66 


02 


8 


53 


46 


84 


21 


58 


95 


31 


67 


02 


7 


54 


47 


84 


22 


59 


95 


32 


67 


03 


6 


55 


47 


85 


22 


59 


96 


32 


68 


03 


5 


56 


48 


86 


23 


60 


96 


33 


69 


04 


4 


57 


49 


86 


24 


60 


97 


33 


69 


05 


3 


58 


49 


87 


24 


61 


98 


34 


70 


05 


2 


59 


9-9998050 


-9998087 


-9998125 


•9998162 


•9998198 


•9998235 


-9998270 


9-9998306 


1 


60 


50 


88 


25 


62 


99 


35 


71 


06 





// 


43' 


42' 


41' 


40' 


39' 


38' 


37' 


36' 


" 



LOS. COSINE 1°. 



Table ii.] log. tan. 88°. 


135 1 


If 


16' , 17' 


18' 


19' 


20' 


21' 


22' 


23' , " 1 





11-5191080 


5233067 


5275462 


5318275 


5361514 


5405186 


5449301 


11-5493869 


60 


1 


1777 


^ 3770 


^ 6173 


-^ 8992 


^ 2238 


- 5918 


5450040 


4615 


59 


2 


2473 


"" 4473 


^ 6883 


^ 9710 


^ 2962 


^ 6649 


^ 0780 


5362 


58 


3 


3170 


5177 


7593 


5320427 


3687 


7381 


-^ 1519 


6109 


57 


4 


3867 


58S0 


8304 


1145 


4412 


8113 


2256 


6856 


56 


5 


4564 


6584 


9014 


1862 


5136 


8845 


2998 


7603 


55 


6 


5261 


7288 


9725 


2580 


5861 


9578 


3738 


8351 


54 


7 


5958 


7992 


5280436 


3298 


6586 


5410310 


4477 


9098 


53 


8 


6655 


8696 


1147 


4016 


7311 


1042 


5217 


9846 


52 


9 


7352 


9400 


1858 


4734 


8037 


1775 


5957 


11-5500593 


51 


10 


8050 


5240104 


2569 


5452 


8762 


2508 


6698 


1341 


50 


11 


8748 


0808 


3280 


6170 


9488 


3240 


7438 


2089 


49 


12 


9445 


1513 


3991 


6889 


5370213 


3973 


8178 


2837 


48 


13 


11-5200143 


2217 


4703 


7607 


0939 


4707 


8919 


3^85 


47 


14 


0841 


2922 


5414 


8326 


1665 


5440 


9660 


4334 


46 


15 


1539 


36-27 


6126 


9045 


2391 


6173 


5460401 


5082 


45 


16 


2237 


4332 


6838 


9764 


3117 


6906 


1141 


5831 


44 


17 


2935 


5037 


7550 


5330483 


3843 


7640 


1883 


6580 


43 


18 


3634 


5742 


8262 


1202 


4569 


8374 


2624 


7328 


42 


19 


4332 


6447 


• 8974 


1921 


5296 


9108 


3365 


8077 


41. 


20 


5031 


7153 


9687 


2640 


6022 


9842 


4107 


8827 


40 


21 


5729 


7858 


5290399 


3360 


6749 


5420576 


4848 


9576 


39 


22 


6428 


8564 


1112 


4079 


7476 


1310 


5590 


11-5510325 


38 


23 


7127 


9270 


1824 


4799 


8203 


2044 


6332 


1075 


37 


24 


7826 


9975 


2537 


5519 


8930 


2779 


7074 


1824 


36 


25 


8525 


5250681 


3250 


6239 


9657 


3513 


7816 


2574 


35 


26 


9225 


1388 


3963 


6959 


5380385 


4248 


8558 


3324 


34 


27 


9924 


2094 


4676 


7679 


1112 


4983 


9300 


4074 


33 


28 


11-5210624 


2800 


5389 


8400 


1840 


5718 


5470043 


4824 


32 


29 


1323 


3506 


6103 


9120 


2567 


6453 


0785 


5574 


31 


30 


2023 


4213 


6816 


9841 


3295 


7188 


1528 


6325 


30 


31 


2723 


4920 


7530 


5340561 


4023 


7923 


2271 


7075 


29 


32 


3423 


5626 


8244 


1282 


4751 


8659 


3014 


7826 


28 


33 


4123 


6333 


8957 


2003 


5479 


9394 


3757 


8577 


27 


34 


4823 


7040 


9671 


2724 


6208 


5430130 


4500 


93-28 


26 


35 


5523 


7747 


5300385 


3445 


6936 


0866 


5243 


11-5520079 


25 


36 


6224 


8455 


1100 


4167 


7664 


1602 


5987 


0830 


24 


37 


6924 


9162 


1814 


4888 


8393 


2338 


6731 


1581 


23 


38 


7625 


9869 


2528 


5610 


9122 


3074 


7474 


2333 


22 


39 


8325 


5260577 


3243 


6331 


9851 


3810 


8218 


3084 


21 


40 


9026 


1285 


3957 


7053 


5390580 


4547 


8962 


3836 


20 


41 


9727 


1992 


4672 


7775 


1309 


5283 


9706 


' 4588 


19 


42 


11-5220428 


2700 


5387 


8497 


2038 


60-20 


5480451 


5340 


18 


43 


1129 


3408 


6102 


9219 


2768 


6757 


1195 


6092 


17 


44 


1831 


4116 


6817 


9941 


3497 


7494 


1939 


6844 


16 


45 


2532 


4825 


7532 


5350664 


4227 


8231 


2684 


7596 


15 


46 


3234 


5533 


8248 


1386 


4957 


8968 


3429 


8349 


14 


47 


3935 


6242 


8963 


2109 


5686 


9705 


4174 


9102 


13 


48 


4637 


6950 


9679 


2832 


6416 


5440442 


4919 


9854 


12 


49 


5339 


7659 


5310395 


3554 


7147 


1180 


5664 


11-5530607 


11 


50 


6041 


8368 


1110 


4277 


7877 


1918 


6409 


1360 


10 


51 


6743 


9077 


1826 


5000 


8607 


2656 


7154 


2113 


9 


52 


7445 


9786 


2542 


5724 


9338 


3393 


7900 


2867 


8 


53 


8147 


5270495 


3259 


6447 


5400068 


4132 


8646 


3620 


7 


54 


8850 


1204 


3975 


7170 


0799 


4870 


9391 


4373 


6 


55 


9552 


1914 


4691 


7894 


1530 


5608 


5490137 


5127 


5 


56 


11-5230255 


2623 


5408 


8618 


2261 


6346 


0883 


5881 


4 


57 


0958 


3333 


6125 


9341 


2992 


7085 


1629 


6635 


3 


58 


1661 


4043 


6841 


5360065 


3723 


7824 


2376 


7389 


2 


59 


2364 


4752 


7558 


0789 


4455 


8562 


3122 


8143 


1 


60 


3067 


5462 


8275 


1514 


5186 


9301 


3869 


8897 





// 


43' 


42' 


41' 


40' 


39' 


38' 


37' 


36' 


" 




LOG. COTAN. V. 


=. I 



136 LOG. SINE 88°. [Ta^»Zen.|| 


" 


24' 


25' 


26' 


27' 


28' 


29' 


30' 


31' 


// 





9-9998306 


-9998342 


-9998376 


•9998411 


-9998445 


-9998478 


•9998512 


9-9998544 


60 


1 


07 


oi 42 


o. 77 


^ 11 


C7. 45 


^ 79 


°^ 12 


45 


59 


2 


08 


43 


77 


12 


_ 46 


79 


13 


46 


58 


3 


08 


43 


78 


12 


46 


80 


13 


46 


57 


4 


09 


44 


79 


13 


47 


80 


14 


47 


56 


5 


09 


44 


79 


13 


47 


81 


14 


47 


55 


6 


10 


45 


80 


14 


48 


82 


15 


48 


54 


7 


11 


46 


80 


15 


49 


82 


15 


48 


53 


8 


11 


46 


81 


15 


49 


83 


16 


49 


52 


9 


12 


47 


81 


16 


50 


83 


16 


49 


51 


10 


9 9998312 


-9998347 


-9998382 


-9998416 


-9998450 


•9998484 


•9998517 


9-9998550 


50 


11 


13 


48 


83 


17 


51 


84 


18 


50 


49 


12 


13 


49 


83 


17 


51 


85 


18 


51 


48 


13 


14 


49 


84 


18 


52 


85 


19 


52 


47 


14 


15 


50 


84 


19 


53 


86 


19 


52 


46 


15 


15 


50 


85 


19 


53 


87 


20 


53 


45 


16 


16 


51 


85 


20 


54 


87 


20 


53 


44 


17 


16 


51 


86 


20 


54 


88 


21 


54 


43 


18 


17 


52 


87 


21 


55 


88 


21 


54 


42 


19 


18 


53 


87 


21 


55 


89 


22 


55 


41 


20 


9-9998318 


•9998353 


•9998388 


-9998422 


•9998456 


-9998489 


-9998523 


9^9998555 


40 


21 


19 


54 


88 


23 


56 


90 


23 


56 


39 


22 


19 


54 


89 


23 


57 


91 


24 


56 


38 


23 


20 


55 


89 


24 


58 


91 


24 


57 


37 


24 


21 


55 


90 


24 


58 


92 


25 


57 


36 


25 


21 


56 


SI 


25 


59 


92 


25 


58 


35 


26 


22 


57 


91 


25 


59 


93 


26 


59 


34 


27 


22 


57 


92 


26 


60 


93 


26 


59 


33 


28 


23 


58 


92 


27 


60 


94 


27 


60 


32 


29 


23 


58 


93 


27 


61 


94 


27 


60 


31 


30 


9-9998324 


•9998359 


•9998393 


-9998428 


•9998461 


-9998495 


•9998528 


9-9998561 


30 


31 


25 


60 


94 


28 


62 


95 


29 


61 


29 


32 


25 


60 


95 


29 


63 


96 


29 


62 


28 


33 


26 


61 


95 


29 


63 


97 


30 


62 


27 


34 


26 


61 


96 


30 


64 


97 


30 


63 


26 


35 


27 


62 


96 


31 


64 


98 


31 


63 


25 


36 


28 


62 


97 


31 


65 


98 


31 


64 


24 


37 


28 


63 


97 


32 


65 


99 


32 


65 


23 


38 


29 


64 


98 


32 


66 


99 


32 


65 


22 


39 


29 


64 


99 


33 


67 


-9998500 


33 


66 


21 


40 


9-9998330 


•9998365 


-9998399 


•9998433 


•9998467 


-9998500 


•9998533 


9-9998566 


20 


41 


30 


65 


-9998400 


34 


68 


01 


34 


67 


19 


42 


31 


66 


00 


34 


68 


02 


35 


67 


18 


43 


32 


66 


01 


35 


69 


02 


35 


68 


17 


44 


32 


67 


01 


36 


69 


03 


36 


68 


16 


45 


33 


68 


02 


36 


70 


03 


36 


69 


15 


46 


33 


68 


03 


37 


70 


04 


37 


69 


14 


47 


34 


69 


03 


37 


71 


04 


37 


70 


13 


48 


35 


69 


04 


38 


72 


05 


38 


70 


12 


49 


35 


70 


04 


38 


72 


05 


38 


71 


11 


50 


9-9998336 


•9998370 


-9998405 


•9998439 


-9998473 


-9998506 


-9998539 


9^9998572 


10 


51 


36 


71 


05 


40 


73 


07 


40 


72 


9 


52 


37 


72 


06 


40 


74 


07 


40 


73 


8 


53 


37 


72 


07 


41 


74 


08 


41 


73 


7 


54 


38 


73 


07 


41 


75 


08 


41 


74 


6 


55 


39 


73 


08 


42 


75 


09 


42 


74 


5 


56 


39 


74 


08 


4?? 


76 


09 


42 


75 


4 


57 


40 


75 


09 


43 


77 


10 


43 


75 


3 


58 


40 


75 


09 


44 


77 


10 


43 


76 


2 


59 


41 


76 


10 


44 


78 


11 


44 


76 


1 


60 


42 


76 


11 


45 


78 


12 


44 


77 





" 


35' 


34' 


33' 


33' 


31' 


30' 


29' 


28' 


// 


LOG. COSINE 1°. || 



Table ii.] log. tan. 88°. 


137 11 


// 


24' 


25' 


26' 


27' 


28' 


29' 


30' 


31' 


- i 





11-5538397 


5584397 


5630378 


5676850 


5723824 


5771310 


5819321 


11-5867868 


60 


1 


9652 


^ 5159 


^ 1148 


^ 7628 


;. 4611 


^ 2106 


5820126 


8682 


59 


2 


U-5540406 


"^ 5922 


^ 1919 


^ 8407 


-" 5398 


"" 2902 


^ 0931 


9496 


58 


3 


1161 


6685 


2690 


9186 


6186 


3698 


^ 1736 


11*5870310 


57 


4 


1916 


7447 


3460 


9966 


6973 


4495 


2541 


1124 


56 


5 


2671 


8210 


4232 


5680745 


7761 


5291 


3346 


1938 


55 


6 


3426 


8973 


5003 


1524 


8549 


6088 


4152 


2753 


54 


7 


4181 


9737 


5774 


2304 


9337 


6884 


4957 


3568 


53 


8 


4937 


5590500 


6545 


3084 


5730125 


7681 


5763 


4382 


52 


9 


5692 


1263 


7317 


3864 


0914 


8478 


6569 


5197 


51 


10 


6448 


2027 


8089 


4644 


1702 


9275 


7375 


6012 


50 


11 


7203 


2791 


8861 


5424 


2491 


5780073 


8181 


6828 


49 


12 


7959 


3554 


9633 


6204 


3280 


0870 


8988 


7643 


48 


13 


8715 


4318 


5640405 


6984 


4068 


1668 


9794 


8459 


47 


14 


9471 


5082 


1177 


7765 


4858 


2466 


5830601 


9274 


46 


! 15 


11-5550228 


5847 


1949 


6546 


5647 


3264 


1407 


11-5880090 


45 


16 


0984 


6611 


2722 


9327 


6436 


4062 


2214 


0906 


44 


17 


1741 


7376 


3494 


5690108 


7226 


4860 


3021 


1722 


43 


18 


2497 


8140 


4267 


0889 


8015 


5658 


3829 


2539 


42 


19 


3254 


8905 


5040 


1670 


8805 


6457 


4636 


3355 


41 


20 


4011 


9670 


5813 


2451 


9595 


7255 


5444 


4172 


40 


21 


4768 


5600435 


6587 


3233 


5740385 


8054 


6251 


4989 


39 


22 


5525 


1200 


7360 


4015 


1175 


8853 


7059 


5806 


38 


23 


6283 


1966 


8133 


4796 


1966 


9652 


7867 


6623 


37 


24 


7040 


2731 


8907 


5578 


2756 


5790451 


8675 


7440 


36 


25 


7798 


3497 


9681 


6361 


3547 


1251 


9483 


8257 


35 


26 


8556 


4262 


5650455 


7143 


4338 


2050 


5840292 


9075 


34 


27 


9313 


5028 


1229 


7925 


5128 


2850 


1100 


9893 


33 


28 


11-5560071 


5794 


2003 


8708 


5920 


3650 


1909 


11-5890710 


32 


29 


0829 


6560 


2777 


9490 


6711 


4450 


2718 


1528 


31 


30 


1588 


7327 


3552 


5700273 


7502 


5250 


3527 


2347 


30 


31 


2346 


8093 


4326 


1056 


8294 


6050 


4336 


3165 


29 


32 


3105 


8860 


5101 


1839 


9085 


6850 


5146 


3983 


28 


33 


3863 


9626 


5876 


2623 


9877 


7651 


5955 


4802 


27 


34 


4622 


5610393 


6651 


3406 


5750669 


8451 


6765 


5621 


26 


35 


5381 


1160 


7426 


4189 


1461 


9252 


7575 


6440 


25 


36 


6140 


1927 


8201 


4973 


2253 


5800053 


8384 


7259 


24 


37 


6899 


2694 


8977 


5757 


3046 


0854 


9195 


8078 


23 


38 


7659 


3462 


9752 


6541 


3838 


1656 


5850005 


8897 


22 


39 


8418 


4229 


5660528 


7325 


4631 


2457 


0815 


9717 


21 


40 


9178 


4997 


1304 


8109 


5424 


3259 


1626 


11-5900537 


20 


41 


9937 


5765 


2080 


8894 


6217 


4060 


2436 


1357 


19 


42 


11-5570697 


6532 


2S56 


9678 


7010 


4862 


3247 


2177 


18 


43 


1457 


7300 


3632 


5710463 


7803 


5664 


4058 


2997 


17 


44 


2217 


8069 


4409 


1248 


8596 


6467 


4869 


3817 


16 


45 


2977 


8837 


5185 


2032 


9390 


7269 


5681 


4638 


15 


46 


3738 


9605 


5962 


2818 


5760184 


8071 


6492 


5458 


14 


47 


4498 


5620374 


6739 


3603 


0977 


8874 


7304 


6279 


13 


48 


5259 


1143 


7516 


4388 


1771 


9677 


8115 


7100 


12 


49 


6020 


1911 


8293 


5174 


2566 


5810480 


8927 


7921 


11 


50 


6781 


2680 


9070 


5959 


3360 


1283 


9739 


8742 


10 


51 


7542 


3450 


9847 


6745 


4154 


2086 


5860552 


9564 


9 


52 


8303 


4219 


5670625 


7531 


4949 


2889 


1364 


11-5910385 


8 


53 


9064 


4988 


1402 


8317 


5743 


3693 


2177 


1207 


7 


54 


9826 


5758 


2180 


9103 


6538 


4496 


2989 


2029 


6 


55 


11*5580587 


6527 


2958 


9890 


7333 


5300 


3802 


2851 


5 


56 


1349 


7297 


3736 


5720676 


8128 


6104 


4615 


3673 


4 


57 


2111 


8067 


4514 


1463 


8924 


6908 


5428 


4495 


3 


58 


2873 


8837 


5293 


2250 


9719 


7712 


6241 


5318 


2 


59 


3635 


9607 


6071 


3037 


5770515 


8517 


7055 


6141 


1 


60 


4397 


5630378 


6850 


3824 


1310 


9321 


7868 


6963 





// 


35' 


Wm 


33' 


32' 


31' 


30' 


29' 


28' 


" 




LOG. COTAN. 1°. 


1 




li 


I* 




' 


R 


■"■'"■"" 









138 LOG. SINE 88°. [Table ii.|| 


'/ 


32' 


33' 


34' 


35' 


36' 


37' 


38' 


39' , " II 





9-9998577 


-9998609 


•9998641 


•9998672 


•9998703 


-9998734 


•9998764 


9-9998794 


60 


1 


77 


^ 10 


o» 41 


<^ 73 


o 04 


°^ 35 


°^ 65 


95 


59 


2 


78 


10 


42 


73 


04 


35 


65 


95 


58 


3 


79 


11 


42 


74 


05 


36 


66 


96 


57 


4 


79 


11 


43 


74 


05 


36 


66 


96 


56 


5 


80 


12 


44 


75 


06 


37 


67 


97 


55 


6 


80 


12 


44 


75 


07 


37 


67 


97 


54 


7 


81 


13 


45 


76 


07 


38 


68 


98 


53 


8 


81 


13 


45 


76 


08 


38 


68 


98 


52 


9 


9-9998582 


•9998614 


•9998646 


•9998677 


■9998708 


•9998739 


•9998769 


9-9998799 


51 


10 


82 


14 


46 


78 


09 


39 


69 


99 


50 


11 


83 


15 


47 


78 


09 


40 


70 


9-9998800 


49 


12 


83 


16 


47 


79 


10 


40 


70 


00 


48 


13 


84 


16 


48 


79 


10 


41 


71 


01 


47 


14 


84 


17 


48 


80 


11 


41 


71 


01 


46 


15 


85 


17 


49 


80 


11 


42 


72 


02 


45 


16 


86 


18 


49 


81 


12 


42 


72 


02 


44 


17 


86 


18 


50 


81 


12 


43 


73 


03 


43 


18 


87 


19 


50 


82 


13 


43 


73 


03 


42 


19 


9-9998587 


•9998619 


•9998651 


•9998682 


•9998713 


•9998744 


•9998774 


9-9998804 


41 


20 


88 


20 


51 


83 


14 


44 


74 


05 


40 


21 


88 


20 


52 


83 


14 


45 


75 


05 


39 


22 


89 


21 


52 


84 


15 


45 


75 


05 


38 


23 


89 


21 


53 


84 


15 


46 


76 


06 


37 


24 


90 


22 


54 


85 


16 


46 


76 


06 


36 


25 


90 


22 


54 


85 


16 


47 


77 


07 


35 


26 


91 


23 


55 


86 


17 


47 


77 


07 


34 


27 


91 


23 


55 


86 


17 


48 


78 


08 


33 


28 


92 


24 


56 


87 


18 


48 


78 


08 


32 


29 


9-9998593 


•9998625 


•9998656 


•9998687 


'9998718 


•9998749 


•9998779 


9^9998809 


31 


30 


93 


25 


57 


88 


19 


49 


79 


09 


30 


31 


94 


26 


57 


88 


19 


50 


80 


10 


29 


32 


94 


26 


58 


89 


20 


50 


80 


10 


28 


33 


95 


27 


58 


89 


20 


51 


81 


11 


27 


34 


95 


27 


59 


90 


21 


51 


81 


11 


26 


35 


96 


28 


59 


90 


21 


52 


82 


12 


25 


36 


96 


28 60 


91 


22 


52 


82 


12 


24 


37 


97 


29 


60 


92 


22 


53 


83 


13 


23 


38 


97 


29 


61 


92 


23 


53 


83 


13 


22 


39 


9-9998598 


•9998630 


•9998661 


•9998693 


•9998723 


•9998754 


•9998784 


9-9998814 


21 


40 


98 


30 


62 


93 


24 


54 


84 


14 


20 


41 


99 


31 


62 


94 


24 


55 


85 


15 


19 


42 


9-9998600 


31 


63 


94 


25 


55 


85 


15 


18 


43 


00 


32 


63 


95 


25 


56 


86 


16 


17 


44 


01 


32 


64 


95 


26 


56 


86 


16 


16 


45 


01 


33 


65 


96 


26 


57 


87 


17 


15 


46 


02 


34 


65 


96 


27 


57 


87 


17 


14 


47 


02 


34 


66 


97 


27 


58 


88 


18 


13 


48 


03 


35 


66 


97 


28 


58 


88 


18 


12 


49 


9-9998603 


•9998635 


•9998667 


•9998698 


•9998728 


•9998759 


•9998789 


9-9998819 


11 


50 


04 


36 


67 


98 


29 


59 


89 


19 


10 


51 


04 


36 


68 


99 


30 


60 


90 


20 


9 


52 


05 


37 


68 


99 


30 


60 


90 


20 


8 


53 


05 


37 


69 


•9998700 


31 


61 


91 


21 


7 


54 


06 


38 


69 


00 


31 


61 


91 


21 


6 


55 


06 


38 


70 


01 


32 


62 


92 


21 


5 


56 


07 


39 


70 


01 


32 


62 


92 


22 


4 


57 


08 


39 


71 


02 


33 


63 


93 


22 


3 


58 


08 


40 


71 


02 


33 


63 


93 


23 


2 


59 


9-9998669 


•9998640 


•9998672 


•9998703 


•9998734 


•9998764 


•9998794 


9-9998823 


1 


60 


09 


41 


72 


03 


34 


64 


94 


24 





// 


2r 


26' 


25' 


24' 


23' 


22' 


• 21' 


SO' " 11 


LOG. COSINE \°. i 



Table II.] LOG. TAN. 88^ 


139 




" 


32' 


33' 


34' 


35' 


36' 37' 


38' 


39' 1 " 







11-5916962 


5966619 


6016848 


606766^ 


6119082 6171114 


6223777 


11-6277085 


60 




1 


7786 


A. 7451 


^ 7690 


^ 85U 


) 9944;. 1986 


^ 4660 


7979 


59 




2 


8609 


^ 8284 


-" 8533 


-" 936C 


6120806 


^ 2859 


^ 5543 


8873 


58 




3 


9433 


9117 


9375 


6070221 


^ 1668 


3732 


6426 


9768 


57 




4 


11-5920256 


9950 


6020218 


107S 


'^ 2531 


4605 


7310 


U -6280662 


56 




5 


1080 


5970783 


1060 


1926 


3394 


5478 


8194 


1557 


55 




6 


1903 


1616 


1903 


277C 


4257 


6352 


9078 


2452 


54 




7 


2727 


2449 


2746 


3632 


5120 


7225 


9962 


3347 


53 




8 


3551 


3283 


3589 


4485 


5983 


8099 


6230847 


4242 


52 




9 


4376 


4116 


4433 


5336 


6847 


8973 


1731 


5138 


51 




10 


5200 


4950 


5276 


6192 


7710 


9847 


2616 


6033 


50 




11 


6025 


5784 


6120 


7045 


8574 


6180721 


3501 


6929 


49 




12 


6849 


6619 


6964 


7899 


9438 


1596 


4386 


7825 


48 




13 


7674 


7453 


7808 


8753 


6130302 


2470 


5271 


8722 


47 




14 


8499 


8287 


8652 


9607 


1167 


3345 


6157 


9618 


46 




15 


9324 


9122 


9497 


6080462 


2031 


4220 


7042 


11-6290515 


45 




16 


11-5930150 


9957 


6030341 


1316 


2896 


5095 


7928 


1411 


44 




17 


0975 


5980792 


1186 


2171 


3761 


5970 


8814 


2308 


43 




18 


1801 


1627 


2031 


3026 


4626 


6846 


9701 


3206 


42 




19 


2626 


2462 


2876 


3881 


5491 


7722 


6240587 


4103 


41 




20 


3452 


3298 


3721 


4736 


6357 


8597 


1473 


5001 


40 




21 


4278 


4133 


4566 


5591 


7222 


9473 


2360 


5898 


39 




22 


5105 


4969 


5412 


6447 


6088 


6190350 


3247 


6796 


38 




23 


5931 


5805 


6258 


7303 


8954 


1226 


4134 


7694 


37 




24 


6758 


6641 


7103 


8158 


9820 


2102 


5021 


8593 


36 




25 


7584 


7477 


7950 


9014 


6140686 


2979 


5909 


9491 


35 




26 


8411 


8314 


8796 


9871 


1552 


3856 


6797 


11-6300390 


34 




27 


9238 


9150 


9642 


6090727 


2419 


4733 


7684 


1289 


33 




28 


11-5940065 


9987 


6040489 


1583 


3286 


5610 


8572 


2188 


32 




29 


0893 


5990824 


1335 


2440 


4153 


6488 


9461 


3087 


31 




30 


1720 


1661 


2182 


3297 


5020 


7366 


6250349 


3986 


30 




31 


2548 


2498 


3029 


4154 


5887 


8243 


1238 


4886 


29 




32 


3376 


3336 


3876 


5011 


6755 


9121 


2126 


5785 


28 




33 


4204 


4173 


4724 


5869 


7622 


9999 


3015 


6665 


27 




34 


5032 


5011 


5571 


6726 


8490 


6200878 


3904 


7586 


26 




35 


5860 


5849 


6419 


7584 


9358 


1756 


4794 


8486 


25 




36 


6689 


6687 


7267 


8442 


6150226 


2635 


5683 


9386 


24 




37 


7517 


7525 


8115 


9300 


1095 


3514 


6573 


11-6310287 


23 




38 


8346 


8363 


8963 


6100158 


1963 


4393 


7462 


1188 


22 




39 


9175 


9202 


9811 


1016 


2832 


5272 


8353 


2089 


21 




40 


11-5950004 


6000041 


6050660 


1875 


3701 


6151 


9243 


2990 


20 




41 


0833 


0879 


1508 


2734 


4570 


7031 


6260133 


3892 


19 




42 


1663 


1718 


2357 


3592 


5439 


7911 


1024 


4793 


18 




43 


2492 


2558 


3206 


4452 


6308 


8791 


1914 


5695 


17 




44 


3322 


3397 


4055 


5311 


7178 


9671 


2805 


6597 


16 




45 


4152 


4236 


4905 


6170 


8047 


6210551 


3696 


7499 


15 




46 


4982 


5076 


5754 


7030 


8917 


1431 


4588 


8402 


14 




47 


5812 


5916 


6604 


7889 


9787 


2312 


5479 


9304 


13 




48 


6642 


6756 


7454 


8749 


6160658 


3193 


6371 


11-6320207 


12 




49 


7473 


7596 


8304 


9609 


1528 


4074 


7263 


1110 


11 




50 


8304 


8436 


9154 


6110470 


2399 


4955 


8155 


2013 


10 




51 


9134 


9277 


6060004 


1330 


3269 


5836 


9047 


2917 


9 




52 


9965 


6010117 


0855 


2191 


4140 


6718 


9939 


3820 


8 




53 


11-5960797 


0958 


1705 


3051 


5011 


7600 


6270832 


4724 


7 




54 


1628 


1799 


2556 


3912 


5883 


8481 


1725 


5628 


6 




55 


, 2459 


2640 


3407 


4773 


6754 


9364 


2617 


6532 


5 




56 


3291 


3481 


4258 


5635 


7626 


6220246 


3511 


7436 


4 




57 


4123 


4323 


5109 


6496 


8497 


1128 


4404 


8340 


3 




58 


4955 


5165 


5961 


7358 


9369 


2011 


5297 


9245 


2 




59 


5787 


6006 


6813 


8220 


6170242 


2894 


6191 


11-6330150 


1 




60 


6619 


6848 7664 9082 


1114 


3777 


7085 


1055 







// 


27 


26' 1 25' 24' 


23' 


22' 


21' 


20' 


" 






LOG. COTAN. 1^. 


^ 





140 LOG. SINE 88°. [Table 


n 


'/ 


40' 


41' 


42' 


43' 


44' 


45' 1 


46' 47' , 


- 





9-9998824 


•9998853 


9998882 


9998911 


9998939 


9998966 • 


9998994 9-9999021 60 11 


1 


24 


<^ 54 


^ 83 


^ 11 


^ 39 


^ 61^ 941 


21 


59 


2 


25 


54 


83 


11 


40 


67 


95 


22 


58 


3 


25 


55 


83 


12 


40 


68 


95 


22 


57 


4 


26 


55 


84 


12 


40 


68 


96 


23 


56 


5 


26 


56 


84 


13 


41 


69 


96 


23 


55 


6 


27 


56 


85 


13 


41 


69 


96 


23 


54 


7 


27 


57 


85 


14 


42 


70 


97 


24 


53 


8 


28 


57 


86 


14 


42 


70 


97 


24 


52 


9 


9-9998828 


•9998858 


9998886 


9998915 


9998943 


9998971 


9998998 


9-9999025 


51 


10 


29 


58 


87 


15 


43 


71 


98 


25 


50 


11 


29 


58 


87 


16 


44 


71 


99 


26 


49 


12 


30 


59 


88 


16 


44 


72 


99 


26 


48 


13 


30 


59 


88 


17 


45 


72 


•9999000 


27 


47 


14 


31 


60 


89 


17 


45 


73 


00 


27 


46 


15 


31 


60 


89 


18 


46 


73 


01 


27 


45 


16 


32 


61 


90 


18 


46 


74 


01 


28 


44 


17 


32 


61 


90 


19 


47 


74 


01 


28 


43 


18 


33 


62 


91 


19 


47 


75 


02 


29 


42 


19 


9-9998833 


-9998862 


•9998891 


•9998919 


-9998947 


•9998975 


•9999002 


9-9999029 


41 


20 


34 


63 


92 


20 


48 


76 


03 


30 


40 


21 


34 


63 


92 


20 


48 


76 


03 


30 


39 


22 


35 


64 


93 


21 


49 


76 


04 


31 


38 


23 


35 


64 


93 


21 


49 


77 


04 


31 


37 


24 


36 


65 


93 


22 


50 


77 


05 


31 


36 


25 


36 


65 


94 


22 


50 


78 


05 


32 


35 


26 


37 


66 


94 


23 


51 


78 


06 


32 


34 


27 


37 


66 


95 


23 


51 


79 


06 


33 


33 


28 


38 


67 


95 


24 


52 


79 


06 


33 


32 


29 


9-9998838 


-9998867 


•9998896 


•9998924 


-9998952 


•9998980 


•9999007 


9-9999034 


31 


30 


39 


68 


96 


25 


53 


80 


07 


34 


30 


31 


39 


68 


97 


25 


53 


81 


08 


35 


29 


32 


40 


69 


97 


26 


53 


81 


08 


35 


28 


33 


40 


69 


98 


26 


54 


81 


09 


35 


27 


34 


41 


70 


98 


26 


54 


82 


09 


36 


26 


35 


41 


70 


99 


27 


55 


82 


10 


36 


25 


36 


42 


71 


99 


27 


55 


63 


10 


37 


24 


37 


42 


71 


•9998900 


28 


56 


83 


10 


37 


23 


38 


42 


71 


00 


28 


56 


84 


11 


38 


22 


39 


9-9998843 


-9998872 


-9998901 


-9998929 


-9998957 


-9998984 


•9999011 


9-9999038 


21 


40 


43 


72 


01 


29 


57 


85 


12 


39 


20 


41 


44 


73 


02 


30 


58 


85 


12 


39 


19 


42 


44 


73 


02 


30 


58 


86 


13 


39 


18 


43 


45 


74 


02 


31 


59 


86 


13 


40 


17 


44 


45 


74 


03 


31 


59 


86 


14 


40 


16 


45 


46 


75 


03 


32 


59 


87 


14 


41 


15 


46 


46 


75* 04 


32 


60 


87 


14 


41 


14 


47 


47 


76r 04 


33 


60 


8E 


15 


42 


13 


48 


47 


76 


05 


33 


61 


8? 


15 


42 


12 


49 


9-9998848 


•9998877 


•9998905 


-9998933 


-9998961 


•999898g 


•9999016 


9-9999043 


11 


'50 


48 


77 


06 


3^ 


62 


8? 


l£ 


43 


10 


51 


49 


78 


oe 


34 


65 


9C 


17 


4S 


a 


52 


4c 


7e 


07 


31 


) 6S 


9( 


17 


U 


8 


53 


5C 


n 


07 


3f 


) 6^ 


i 9] 


1£ 


A4 


7 


54 


5C 


) 7c 


-Of 


] 3( 


) 64 


\ 9] 


u 


4J 


6 


55 


5] 


8C 


Qi 


I 3( 


5 64 


I 9] 


L li 


4f 


5 


56 


5 


L 8( 


OJ 


i 3' 


f 61 


) 95 


I li 


) 4( 


) 4 


57 


5^ 


I 8 


0^ 


) 3' 


1 61 


) 95 


I V 


) 4( 


5 3 


58 


5' 


I 8] 


K 


) 3 


3 6. 


J 9 


J 2( 


) 4' 


r 2 


59 


9-999885 


? -999888' 


I ^999891 


3 '999893 


3 •999896( 


3 -999899 


J •999902( 


) 9-999904' 


r 1 


60 


5 


I 8 


I 1 


I 3 


9 6 


3 9 


1 2 


I 4' 


r 


" 


19' 


18' 


17' 1 16' 


15' 


14' 


13' 


12' 


// 


LOG. COSINE 1°. 1| 



Table ii.] log. tan. 88°. 


-ll 


// 


40' 


41' 


42' 


43' 


44' 


45' 


46' 


47' 


// 





11-6331055 


6385703 


6441047 


6497105 


6553895 


6611437 


6669751 


11-6728857 


60 


1 


1960 


.1. 6620 


^ 1976 


^ 8046 


^ 4848 


^ 2403 


6670729 


9849 


59 


2 


2865 


-^ 7537 


^ 2904 


^ 8986 


•^ 5801 


^ 3368 


^ 1708 


11-6730842 


58 


3 


3771 


8454 


3833 


9927 


6754 


4334 


^ 2687 


1834 


57 


4 


4677 


9371 


4762 


6500868 


7708 


5300 


3666 


2827 


56 


5 


5583 


6390289 


5691 


1809 


8661 


6267 


4646 


3819 


55 


6 


6489 


1206 


6621 


2751 


9615 


7233 


5625 


4812 


54 


7 


7395 


2124 


7550 


3693 


6560569 


8200 


6605 


5806 


53 


8 


8302 


3042 


8480 


4635 


1524 


9167 


7585 


6799 


52 


9 


9208 


3960 


9410 


5677 


2478 


6620134 


8566 


7793 


51 


10 


11-6340115 


4879 


6450340 


6519 


3433 


1102 


9546 


8787 


50 


11 


1022 


5797 


1271 


7461 


4388 


2070 


6680527 


9781 


49 


12 


1930 


6716 


2201 


8404 


5343 


3037 


1508 


11-6740776 


48 


13 


2837 


7635 


3132 


9347 


6298 


4006 


2489 


1770 


47 


14 


3745 


8554 


4063 


6510290 


7254 


4974 


3471 


2765 


46 


15 


4653 


9473 


4994 


1233 


8209 


5942 


4452 


3760 


45 


16 


5561 


6400393 


5926 


2177 


9165 


6911 


5434 


4756 


44 


17 


6469 


1313 


6857 


3121 


6570122 


7880 


6416 


5751 


43 


18 


7377 


2233 


7789 


4064 


1078 


8849 


7399 


6747 


42 


19 


8286 


3153 


8721 


5009 


2035 


9819 


8381 


7743 


41 


20 


9195 


4073 


9653 


5953 


2991 


6630788 


9364 


8740 


40 


21 


11-6350104 


4994 


6460586 


6897 


3948 


1758 


6690347 


9736 


39 


22 


1013 


5914 


1518 


7842 


4906 


2728 


1330 


11-6750733 


38 


23 


1922 


6835 


2451 


8787 


5863 


3698 


2313 


1730 


37 


24 


2832 


7757 


3384 


9732 


6821 


4669 


3297 


2727 


36 


25 


3742 


8678 


4317 


3520678 


7779 


5639 


4281 


3724 


35 


26 


4652 


9599 


5250 


1623 


8737 


6610 


5265 


4722 


34 


27 


5562 


6410521 


6184 


2569 


9695 


7581 


6249 


5720 


33 


28 


6472 


1443 


7118 


3515 


6580653 


8553 


7234 


6718 


32 


29 


7383 


2365 


8052 


4461 


1612 


9524 


8219 


7716 


31 


30 


8293 


3287 


8986 


5408 


2571 


6640496 


9204 


8715 


30 


31 


9204 


4210 


9920 


6354 


3530 


1468 


6700189 


9714 


29 


32 


11-6360115 


5132 


6470855 


7301 


4489 


2440 


1174 


11-6760713 


28 


33 


1026 


6055 


1790 


8248 


5449 


3413 


2160 


1712 


27 


34 


1938 


6978 


2725 


9195 


6409 


4385 


3146 


2711 


26 


35 


2850 


7902 


3660 


6530143 


7369 


5358 


4132 


3711 


25 


36 


3761 


8825 


4595 


1090 


8329 


6331 


5118 


4711 


24 


37 


4673 


9749 


5531 


2038 


9289 


7305 


6105 


5711 


23 


38 


5586 


6420673 


6467 


2986 


6590250 


8278 


7092 


6712 


22 


39 


6498 


1597 


7403 


3934 


1211 


9252 


8079 


7712 


21 


40 


7411 


2521 


8339 


4883 


2172 


6650226 


9066 


8713 


20 


41 


8324 


3445 


9275 


5831 


3133 


1200 


6710053 


9714 


19 


42 


9237 


4370 


6480212 


6780 


4094 


2174 


1041 


11-6770715 


18 


43 


11-6370150 


5295 


1149 


7729 


5056 


3149 


2029 


1717 


17 


44 


1063 


6220 


2086 


8679 


6018 


4124 


3017 


2719 


16 


45 


1977 


7145 


3023 


9628 


6980 


5099 


4005 


3721 


15 


46 


2890 


8071 


3960 


3540578 


7942 


6074 


4994 


4723 


14 


47 


3804 


8996 


4898 


1528 


8905 


7050 


5983 


5726 


13 


48 


4719 


9922 


5836 


2478 


9868 


8025 


6972 


6728 


12 


49 


5633 


6430848 


6774 


3428 


6600831 


9001 


7961 


7731 


11 


50 


6547 


1774 


7712 


4379 


1794 


9977 


8950 


8734 


10 


51 


7462 


2701 


8650 


5329 


2757 


6660954 


9940 


9738 


9 


52 


8377 


3627 


9589 


6280 


3721 


1930 


6720930 


11-6780741 


8 


53 


9292 


4564 


6490528 


7231 


4684 


2907 


1920 


1745 


7 


54 


11-6380207 


5481 


1467 


8183 


5649 


3884 


2910 


2749 


6 


55 


1123 


6403 


2406 


9134 


6613 


4861 


3901 


3754 


5 


56 


2039 


7335 


3345 


6550086 


7577 


5839 


4892 


4758 


4 


57 


2955 


8263 


4285 


1038 


8542 


6816 


5883 


5763 


3 


58 


3871 


9191 


5225 


1990 


9507 


7794 


6874 


6768 


2 


59 


4787 


6440119 


6165 


2943 


6610472 


8772 


7866 


7773 


1 


60 


5703 


1047 


7105 


3895 


1437 


9751 


8857 


8779 





" 


19' 


18' 


17' 


16' 


15' 


14' 


13' 


12' 


" 


Ls 


LOG. COTAN. 1°. 


J| 



142 LOG. SINE 88°. [Table u. 




// 


48' 


49' 


50' 


51' 


52' 


53' 


54' 


55' 


// 







9-9999047 


■9999074 


•9999100 


■9999125 


•9999150 


•9999175 


•9999200 


9^9999224 


60 




1 


48 


^ 74 


^ 00 


^ 26 


^ 51 


^ 76 


^ 00 


24 


59 




2 


49 


75 


00 


26 


51 


76 


00 


24 


58 




3 


49 


75 


01 


26 


52 


76 


01 


25 


57 




4 


49 


75 


01 


27 


52 


77 


01 


25 


56 




5 


50 


76 


02 


27 


52 


77 


02 


26 


55 




6 


50 


76 


02 


28 


53 


77 


02 


26 


54 




7 


51 


77 


03 


28 


53 


78 


02 


26 


53 




8 


51 


77 


03 


29 


54 


78 


03 


27 


52 




9 


9-9999051 


•9999078 


-9999103 


-9999129 


•9999154 


•9999179 


•9999203 


9^9999227 


51 




10 


52 


78 


04 


29 


54 


79 


04 


28 


50 




11 


52 


7S 


04 


30 


55 


80 


04 


28 


49 




12 


53 


79 


05 


30 


55 


80 


04 


28 


48 




13 


53 


79 


05 


31 


56 


80 


05 


29 


47 




14 


54 


80 


06 


31 


56 


81 


05 


29 


46 




15 


54 


80 


06 


31 


57 


81 


06 


30 


45 




16 


54 


81 


06 


32 


57 


82 


06 


30 


44 




17 


55 


81 


07 


32 


57 


82 


06 


30 


43 




18 


55 


82 


07 


33 


58 


82 


07 


31 


42 




19 


9-9999056 


-9999082 


•9999108 


•9999133 


•9999158 


•9999183 


•9999207 


9-9999231 


41 




20 


56 


82 


08 


34 


59 


83 


08 


32 


40 




21 


57 


83 


09 


34 


59 


84 


08 


32 


39 




22 


57 


83 


09 


34 


59 


84 


08 


32 


38 




23 


58 


84 


09 


35 


60 


85 


09 


33 


37 




24 


58 


84 


10 


35 


60 


85 


09 


33 


36 




25 


58 


85 


10 


36 


61 


85 


10 


34 


35 




26 


59 


85 


11 


36 


61 


86 


10 


34 


34 




27 


59 


85 


11 


37 


62 


86 


10 


35 


33 




28 


60 


86 


12 


37 


62 


87 


11 


35 


32 




29 


9-9999060 


•9999086 


-9999112 


•9999137 


•9999162 


■9999187 


•9999211 


9-9999235 


31 




30 


61 


87 


12 


38 


63 


87 


12 


36 


30 




31 


61 


87 


13 


38 


63 


88 


12 


36 


29 




32 


61 


88 


13 


39 


64 


88 


12 


36 


28 




33 


62 


88 


14 


39 


64 


89 


13 


37 


27 




34 


62 


88 


14 


39 


64 


89 


13 


37 


26 




35 


63 


89 


15 


40 


65 


89 


14 


38 


25 




36 


63 


89 


15 


40 


65 


90 


14 


38 


24 




37 


64 


90 


15 


41 


66 


90 


14 


38 


23 




38 


64 


90 


16 


41 


66 


91 


15 


39 


22 




39 


9-9999065 


-9999091 


•9999116 


•9999142 


■9999166 


•9999191 


•9999215 


9-9999239 


21 




40 


65 


91 


17 


42 


67 


91 


16 


39 


20 




41 


65 


91 


17 


48 


67 


92 


16 


40 


19 




42 


66 


92 


18 


43 


68 
68 


92 


16 


40 


18 




43 


66 


92 


18 


43 


93 


17 


41 


17 




44 


67 


93 


18 


44 


69 


93 


17 


41 


16 




45 


67 


93 


19 


44 


69 


93 


18 


41 


15 




46 


68 


94 


19 


44 


69 


94 


18 


42 


14 




47 


68 


94 


20 


45 


70 


94 


18 


42 


13 




48 


68 


94 


20 


45 


70 


95 


19 


43 


12 




49 


9-9999069 


-9999095 


•9999120 


•9999146 


•9999171 


-9999195 


•9999219 


9^9999243 


11 




50 


69 


95 


21 


46 


71 


96 


20 


43 


10 




51 


70 


96 


21 


47 


71 


96 


20 


44 


9 




52 


70 


96 


22 


47 


72 


96 


20 


44 


8 




53 


71 


97 


22 


47 


72 


97 


21 


45 


7 




54 


71 


97 


23 


48 


73 


97 


21 


45 


6 




55 


72 


97 


23 


48 


73 


98 


22 


45 


5 




56 


72 


98 


23 


49 


73 


98 


22 


46 


4 




57 


72 


98 


•24 


49 


74 


98 


22 


46 


3 




58 


73 


99 


24 


49 


74 


99 


23 


47 


2 




59 


9-9999073 


-9999099 


•9999125 


•9999150 


•9999175 


•9999199 


•9999223 


9-9999247 


1 




60 


74 


-9999100 


25 


50 


75 


-9999200 


24 


47 







// 


IV 1 10' 


9' 


8' 


7' 


6' 


5' 


4' 


'' 




LOG. -COSINE 1°. 1 





Tahie 11.] LOG. TAN. 88°. 143 1| 


// 


48' 


49' 


50' 


51' 


52' 


53' 


54' 


55' 1 " 





11-6788779 


6849538 


6911158 


697366E 


703708: 


{7101441 


7166766 


11-72330881 60 


1 


9785 


6850558 


- 2193 


^ m4 


^ 814E 


l^ 2522 


^ 7863 


42021 59 


2 


11-6790790 


- 1578 


'^ 3227 


^ 5764 


"^ 92i: 


J"" 3603 


^ 8960 


5316 58 


3 


1797 


■" 2598 


4262 


6814 


704027^ 


4684 


7170058 


6430 57 


4 


2803 


3619 


5297 


7864 


134^ 


5765 


1156 


7545 56 


5 


3810 


4640 


6333 


89U 


24 IC 


6847 


2254 


8660 55 


6 


4817 


5661 


7369 


9965 


3476 


7929 


3353 


9776 54 


7 


5824 


6682 


8404 


6981016 


454S 


9012 


4451 


11-72408921 53 


8 


6831 


7704 


9441 


2067 


560C 


7110094 


5550 


20081 52 


9 


7839 


8725 


6920477 


3119 


6676 


1177 


6650 


3124 51 


10 


8846 


9747 


1514 


4170 


7744 


2260 


7749 


4240 


50 


11 


9855 


6860770 


2551 


5222 


8811 


3344 


8849 


5357 


49 


12 


11-6800863 


1792 


3588 


6275 


9879 


4428 


9949 


6474 


48 


13 


1871 


2815 


4625 


7327 


7050947 


5512 


7181050 


7592 


47 


14 


2880 


3838 


5663 


8380 


2015 


6596 


2151 


8709 


46 


15 


3889 


4861 


6701 


9433 


3084 


7680 


3252 


9827 f 45 


16 


4898 


5885 


7739 


6990486 


4152 


8765 


4353 


11-7250946 44 


17 


5908 


6908 


8777 


1540 


5221 


9850 


5455 


2064 43 


18 


6917 


7932 


9816 


2593 


6291 


7120935 


6556 


3183 42 


19 


7927 


8957 


6930855 


3647 


7360 


2021 


7658 


4302, 41 


20 


8938 


9981 


1894 


4702 


8430 


3107 


8761 


5422 


40 


21 


9948 


6871006 


2933 


5756 


9500 


4193 


9864 


6542 


39 


22 


11-6810959 


2031 


3973 


6811 


7060571 


5280 


7190966 


7662 


38 


23 


1969 


3056 


5013 


7866 


1641 


6366 


2070 


8782 


37 


24 


2981 


4081 


6053 


8921 


2712 


7453 


3173 


9902 


36 


25 


3992 


5107 


7093 


9977 


3783 


8540 


4277 


11-7261023 


35 


26 


5003 


6133 


8134 


7001033 


4855 


9628 


5381 


2144 34 


27 


6015 


7159 


9175 


2089 


5926 


7130716 


6485 


3266 33 


28 


7027 


8185 


6940216 


3145 


6998 


1804 


7590 


43881 32 


29 


8040 


9212 


1257 


4201 


8070 


2892 


8695 


5510 31 


30 


9052 


6880239 


2299 


5256 


9143 


3981 


9800 


66321 30 


31 


11-6820065 


1266 


3341 


6315 


7070216 


5069 


7200906 


7755 


29 


32 


1078 


2593 


4383 


7373 


1289 


6159 


2012 


8878 


28 


33 


2091 


3321 


5425 


8430 


2362 


7248 


3118 


11-7270001 


27 


34 


3105 


4349 


6468 


948S 


3435 


8338 


4224 


1124 


26 


35 


4118 


5377 


7511 


7010546 


450S 


9428 


5330 


2248 


25 


36 


5132 


6405 


8554 


1605 


5583 


7140518 


6437 


3372 


24 


37 


6146 


7434 


9597 


2663 


665S 


1608 


7545 


4496 


23 


38 


7161 


8462 


6950641 


3722 


7732 


2699 


8652 


562] 


22 


39 


8175 


9492 


1685 


4781 


8807 


3790 


9760 


6746 


21 


40 


9190 


6890521 


2729 


5841 


9882 


4882 


7210868 


7871 


20 


41 


11-6830205 


1560 


3774 


6900 


7080958 


5973 


1976 


8997 


19 


42 


1221 


2580 


4818 


7960 


2033 


7065 


3085 


11-7280123 


18 


43 


2236 


3610 


5863 


9020 


3109 


8157 


4194 


1249 


17 


44 


3252 


4640 


6908 


7020081 


4185 


9250 


5303 


2375 


16 


45 


4268 


5671 


7954 


1142 


5262 


7150342 


6412 


3502 


15 


46 


5285 


6702 


8999 


2203 


6339 


1435 


7522 


4629 


14 


47 


6301 


7733 


6960045 


3264 


7416 


2529 


8632 


5756 


13 


48 


7318 


8764 


1091 


4325 


8493 


3622 


9742 


6884 


12 


49 


8335 


9795 


2138 


5387 


9570 


4716 


7220853 


8011 


11 


50 


9352 


6900827 


3184 


6449 


7090648 


5810 


1964 


9140 


10 


51 


11-6840370 


1859 


4231 


7511 


1726 


6904 


3075 


11-7290268 


9 


52 


1387 


2891 


5278 


8574 


2805 


7999 


4186 


1397 


8 


53 


2405 


3924 


6326 


9637 


3883 


9094 


5298 


2526 


7 


54 


sm 


4957 


7373 


7030700 


4962 


7160189 


6410 


3655 


6 


55 


4442 


5990 


8421 


1763 


6041 


1284 


7522 


4785 


5 


56 


5461 


7023 


9469 


2826 


7121 


2380 


8635 


5915 


4 


57 


6480 


8056 


6970518 


3890 


8200 


3476 


9747 


7045 


3 


58 


7499 


9090 


1567 


4954 


9280 


4572 


r230861 


8175 


2 


59 


8518 


S910124 


2615 


6019 


7100360 


5669 


1974 


9306 


1 


60 


9538 


1158 


3665 


7083 


1441 


6766 


3088 


11-7300437 





// 


W 


10' 


9' 


8' 


7' 


6' 


5' 


4' 


" 




LOG. COTAN. 1°. 11 



144 LOG. SINE 88°. LOG. SINE 89°. [Table ii. 




// 


56' 


57' 


58' 


59' 


0' 


1' 


2' 3' 1 " 1 







9-9999247 


•9999271 


-9999294 


-9999316 


•9999338 


•9999360 


•9999382 


9-9999403 


60 




1 


48 


c 71 


a. 94 


c» 17 


°* 39 


=^ 61 


=^ 82 


03 


59 




2 


48 


71 


94 


17 


39 


61 


83 


04 


58 




3 


49 


72 


95 


17 


40 


61 


83 


04 


57 




4 


49 


72 


95 


18 


40 


62 


83 


04 


56 




5 


49 


73 


96 


18 


40 


62 


84 


05 


55 




6 


50 


73 


96 


19 


41 


63 


84 


05 


54 




7 


50 


73 


96 


19 


41 


63 


84 


05 


53 




8 


50 


74 


97 


19 


41 


.63 


85 


06 


52 




9 


9-9999251 


•9999274 


-9999297 


•9999320 


■9999342 


•9999364 


•9999385 


9-9999406 


51 




10 


51 


75 


97 


20 


42 


64 


85 


06 


50 




11 


52 


75 


98 


20 


43 


64 


86 


07 


49 




12 


52 


75 


98 


21 


43 


65 


86 


07 


48 




13 


52 


76 


99 


21 


43 


65 


86 


08 47 1 




14 1 


53 


76 


99 


21 


44 


65 


87 


08 


46 




15 


53 


76 


99 


22 


44 


66 


87 


08 


45 




16 


54 


77 


•9999300 


22 


44 


66 


88 


09 


44 




17 


54 


77 


00 


23 


45 


66 


88 


09 


43 




181 


54 


78 


00 


23 


45 


67 


88 


09 


42 




19 


9-9999255 


9999278 


-9999301 


9999323 


•9999345 


•9999367 


•9999389 


9^9999410 


41 




20 


55 


78 


01 


24 


46 


68 


89 


10 


40 




21 ' 


56 


79 


02 


24 


46 


68 


89 


10 


39 




22 


56 


79 


02 


24 


47 


68 


90 


11 


38 




23 


56 


80 


02 


25 


47 


69 


90 


11 


37 




24 


57 


80 


03 


25 


47 


69 


90 


11 


36 




25 


57 


80 


03 


26 


48 


69 


91 


12 


35 




26 


58 


81 


04 


26 


48 


70 


91 


12 


34 




27 


58 


81 


04 


26 


48 


70 


91 


12 


33 




28 


58 


81 


04 


27 


49 


70 


92 


13 


32 




29 


9-9999259 


-9999282 


-9999305 


•9999327 


-9999349 


-9999371 


•9999392 


9^9999413 


31 




30 


59 


82 


05 


27 


49 


71 


92 


13 


30 




31 


59 


83 


05 


28 


50 


72 


93 


14 


29 




32 


60 


83 


06 


28 


50 


72 


93 


14 


28 




33 


60 


83 


06 


29 


51 


72 


94 


14 


27 




34 


61 


84 


07 


29 


51 


73 


94 


15 


26 




35 


61 


84 


07 


29 


51 


73 


94 


15 


25 




36 


61 


85 


07 


30 


52 


73 


95 


16 


24 




37 


62 


85 


08 


30 


52 


74 


95 


16 


23 




38 


62 


85 


08 


30 


52 


74 


95 


16 


22 




39 


9-9999263 


•9999286 


•9999308 


•9999331 


-9999353 


•9999374 


-9999396 


9^9999417 


21 




40 


63 


86 


09 


31 


53 


75 


96 


17 


20 




41 


63 


86 


09 


31 


53 


75 


96 


17 


19 




42 


64 


87 


10 


32 


54 


75 


97 


18 


18 




43 


64 


87 


10 


32 


54 


76 


97 


18 


17 




44 


65 


88 


10 


33 


55 


76 


97 


18 


16 




45 


65 


88 


11 


33 


55 


77 


98 


19 


15 




46 


65 


88 


11 


33 


55 


77 


98 


19 


14 




47 


66 


89 


11 


34 


56 


77 


98 


19 


13 




48 


66 


89 


12 


34 


56 


78 


99 


20 


12 




49 


9-9999266 


-9999289 


-9999312 


•9999334 


•9999356 


-9999378 


-9999399 


9^9999420 


11 




50 


67 


90 


13 


35 


57 


78 


•9999400 


20 


10 




51 


67 


90 


13 


35 


57 


79 


00 


21 


9 




52 


68 


91 


13 


36 


57 


79 


00 


21 


8 




53 


68 


91 


14 


36 


58 


79 


01 


21 


7 




54 


68 


91 


14 


36 


58 


80 


01 


22 


6 




55 


69 


92 


14 


37 


59 


80 


01 


22 


5 




56 


69 


92 


15 


37 


59 


80 


02 


22 


4 




57 


70 


93 


15 


37 


59 


81 


02 


23 


3 




58 


70 


93 


16 


38 


60 


81 


02 


23 


2 




59 


9-9999270 


-9999293 


■9999316 


•9999338 


•9999360 


•9999382 


-9999403 


9^9999423 


1 




60 


71 


94 


16 


38 


60 


82 


03 


24 







'' 


3' 


2' 


1' 


0' 


59' 


58' 


57' 


56' 


" 




LOG. COSINE 1°. LOG. COSINE 0°. 





Table n.] log. tan. 88°. log. tan. 89°. 145]] 


// 


56' 


bT 


58' 


59' 


0' 


1' 


2' 


3' 


// 





11-7300437 


7368847 


7438351 


7508985 


7580785 


7653792 


7728047 


11-7803592 


60 


1 


1569 


9996 


9519 


7510172 


^ 1992 


^ 5020 


9295 


4863 


59 


2 


2700 


7371146 


7440667 


^ 1359 


^ 3199 


^ 6247 


7730544 


6133 


58 


3 


3832 


- 2296 


^ 1856 


*" 2547 


4407 


7475 


^ 1793 


7404 


57 


4 


4965 


^ 3446 


^ 3024 


3735 


5614 


8703 


^ 3043 


8676 


56 


5 


6097 


4597 


4194 


4923 


6823 


9932 


4292 


9947 


55 


6 


7230 


5748 


5363 


6112 


8031 


7661161 


5543 


11-781122C 


54 


7 


8363 


6899 


6533 


7301 


9240 


2390 


6793 


2492 


53 


8 


9497 


8050 


7703 


8490 


7590449 


3620 


8044 


376£ 


52 


9 


11-7310630 


9202 


8873 


9679 


1658 


4850 


9295 


503£ 


51 


10 


1764 


7380354 


7450044 


7520869 


2868 


6080 


7740547 


6312 


50 


1 11 


2899 


1507 


1215 


2060 


4078 


7311 


1799 


7586 


49 


12 


4033 


2659 


2386 


3250 


5289 


8542 


3051 


8860 


48 


13 


5168 


3812 


3558 


4441 


6500 


9773 


4304 


11-7820135 


47 


14 


6304 


4966 


4730 


5632 


7711 


7671005 


5557 


1410 


46 


15 


7439 


6119 


5902 


6824 


8922 


2237 


6810 


2686 


45 


16 


8575 


7273 


7075 


8016 


7600134 


3470 


8064 


3962 


44 


17 


9711 


8427 


8248 


9208 


1346 


4703 


9318 


523g 


43 


18 


11-7320847 


9582 


9421 


7530401 


2559 


5936 


7750573 


6514 


42 


19 


1984 


7390737 


7460594 


1593 


3772 


7169 


1828 


7791 


41 


20 


3121 


1892 


1768 


2787 


4985 


8403 


3083 


9069 


40 


21 


4258 


3047 


2942 


3980 


6198 


9637 


4339 


11-7830347 


39 


22 


5396 


4203 


4117 


5174 


7412 


7680872 


5595 


1625 


38 


23 


6534 


5359 


5292 


6368 


8627 


2107 


6851 


2903 


37 


24 


7672 


6515 


6467 


7563 


9841 


3342 


8108 


4182 


36 


25 


8811 


7672 


7642 


8758 


7611056 


4578 


9365 


5461 


35 


26 


9949 


8829 


8818 


9953 


2271 


5814 


7760622 


6741 


34 


27 


11-7331089 


9986 


9994 


7541148 


3487 


7050 


1880 


8021 


33 


28 


2228 


7401144 


7471171 


2344 


4703 


8287 


3138 


9301 


32 


29 


3368 


2302 


2347 


3540 


5919 


9524 


4396 


11-7840582 


31 


30 


4508 


3460 


3524 


4737 


7135 


7690761 


5655 


1863 


30 


31 


5648 


4619 


4702 


5934 


8352 


1999 


6915 


3145 


29 


32 


6788 


5777 


5879 


7131 


9570 


3237 


8174 


4427 


28 


33 


7929 


6937 


7057 


8328 


7620787 


4475 


9434 


5709 


27 


34 


9071 


8096 


8236 


9526 


2005 


5714 


7770695 


6992 


26 


35 


11-7340212 


9256 


9414 


7550724 


3224 


6953 


1955 


8275 


25 


36 


1354 


7410416 


7480593 


1923 


4442 


8193 


3216 


9558 


24 


37 


2496 


1576 


1773 


3121 


5661 


9432 


4478 


11-7850842 


23 


38 


3638 


2737 


2952 


4320 


6880 


7700673 


5740 


2126 


22 


39 


4781 


3898 


4132 


5520 


8100 


1913 


7002 


3411 


21 


40 


5924 


5059 


5312 


6720 


9320 


3154 


8264 


4696 


20 


41 


7067 


6221 


6493 


7920 


7630540 


4395 


9527 


5981 


19 


42 


8211 


7383 


7674 


9120 


1761 


5637 


7780790 


7267 


18 


43 


9355 


8545 


8855 


7560321 


2982 


6879 


2054 


8553 


17 


44 


11-7350499 


9708 


7490036 


1522 


4204 


8121 


3318 


9839 


16 


45 


1643 


7420871 


1218 


2724 


5425 


9364 


4582 


11-7861126 


15 


46 


2788 


2034 


2400 


3925 


6647 


7710607 


5847 


2413 


14 


47 


3933 


3197 


3583 


5128 


7870 


1850 


7112 


3701 


13 


48 


5079 


4361 


4766 


6330 


9092 


3094 


8378 


4989 


12 


49 


6224 


5525 


5949 


7533 


7640316 


4338 


9644 


6277 


11 


50 


7370 


6690 


7132 


8736 


1539 


5583 


7790910 


7566 


10 1 


51 


8517 


7854 


8316 


9939 


2763 


6827 


2176 


8855 


9 


52 


9663 


9019 


9500 


7571143 


3987 


8073 


3443 


11-7870145 


8 


53 


11-7360810 


7430185 


7500685 


2347 


5211 


9318 


4711 


1434 


7 


54 


1957 


1350 


1869 


3552 


6436 


7720564 


5978 


2725 


6 


55 


3105 


2516 


3054 


4756 


7661 


1810 


7246 


4015 


5 


56 


4253 


3683 


4240 


5962 


8887 


3057 


8515 


5306 


4 


57 


5401 


4849 


5426 


7167 


7650113 


4304 


9784 


6598 


3 


58 


6549 


6016 


6612 


8373 1339 


5551 


7801053 


7890 


2 


59 


7698 


7183 


7798 


9579 2565 


6799 


2322 


9182 


1 


60 


8847 


8351 89851 


7580785 3792 


8047 


3592 


11-7880474 





"1 


3' 


2' r 1 0' 1 59' 1 


58' 


57' 


56' 


" 




LOG. COTAN. 1°. LOG. COTAN. 0^. (J 



13 



146 LOG. SINE 89^ [ToJ^e II. il 


// 


4' 


5' 


6' 


7' 8' 9' , 10' • 


11' 1 


"1 





9-9999424 


9999444 


9999464 


9999484 -9999503 -9999522-9999541 9-99995591 


60 


1 


24 


3s 44 


^ 65 


5» 84 «* 03 o» 22 


^ .41 


59 


59 


2 


24 


45 


65 


84 


04 


23 


41 


59 


58 


3 


25 


45 


65 


85 


04 


23 


42 


60 


57 


4 


25 


46 


66 


85 


04 


23 


42 


60 


56 


5 


25 


46 


66 


85 


05 


24 


42 


60 


55 


6 


26 


46 


66 


86 


05 


24 


42 


61 


54 


7 


26 


47 


67 


86 


05 


24 


43 


61 


53 


3 


27 


47 


67 


86 


06 


25 


43 


61 


52 


9 


9-9999427 


9999447 


9999467 


•9999487 


9999506 


9999525 


•9999543 


9^9999562 


51 


10 


27 


48 


67 


87 


06 


25 


44 


62 


50 


11 


28 


48 


68 


87 


07 


26 


44 


62 


49 


12 


28 


48 


68 


88 


07 


26 


44 


62 


48 


13 


28 


49 


68 


88 


07 


26 


45 


63 


47 


14 


29 


49 


69 


88 


08 


26 


45 


63 


46 


15 


29 


49 


69 


89 


08 


27 


45 


63 


45 


16 


29 


50 


69 


89 


08 


27 


46 


64 


44 


17 


30 


50 


70 


89 


09 


27 


46 


64 


43 


18 


30 


50 


70 


90 


09 


28 


46 


64 


42 


19 


9-9999430 


9999451 


-9999470 


-9999490 


-9999509 


•9999528 


•9999546 


9-9999565 


41 


20 


31 


51 


71 


90 


09 


28 


47 


65 


40 


21 


31 


51 


71 


91 


10 


29 


47 


65 


39 


22 


31 


52 


71 


91 


10 


29 


47 


65 


38 


23 


32 


52 


72 


91 


10 


29 


48 


66 


37 


24 


32 


52 


72 


92 


11 


30 


48 


66 


36 


25 


32 


53 


72 


92 


11 


30 


48 


66 


35 


26 


33 


53 


73 


92 


11 


30 


49 


67 


34 


27 


33 


53 


73 


93 


12 


30 


49 


67 


33 


28 


33 


54 


73 


93 


12 


31 


49 


67 


32 


29 


9-9999434 


•9999454 


-9999474 


-9999493 


-9999512 


•9999531 


-9999549 


9-9999567 


31 


30 


34 


54 


74 


94 


13 


31 


50 


68 


30 


31 


34 


55 


74 


94 


13 


32 


50 


68 


29 


32 


35 


55 


75 


94 


13 


32 


50 


68 


28 


33 


35 


65 


75 


95 


14 


32 


51 


69 


27 


34 


35 


56 


75 


95 


14 


33 


51 


69 


26 


35 


36 


56 


76 


95 


14 


33 


51 


69 


25 


36 


36 


56 


76 


95 


15 


33 


52 


70 


24 


37 


36 


57 


76 


96 


15 


34 


52 


70 


23 


38 


37 


57 


77 


96 


15 


34 


52 


70 


22 


39 


9-9999437 


-9999457 


-9999477 


-9999496 


-9999515 


-9999534 


•9999552 


9^9999570 


21 


40 


37 


58 


77 


97 


16 


34 


53 


71 


20 


41 


38 


58 


78 


97 


16 


35 


53 


71 


19 


42 


38 


58 


78 


97 


16 


35 


53 


71 


18 


43 


38 


59 


78 


98 


17 


35 


54 


72 


17 


44 


39 


59 


79 


98 


17 


36 


54 


72 


16 


45 


39 


59 


79 


98 


17 


36 


54 


72 


15 


46 


39 


60 


79 


99 


18 


36 


55 


73 


14 


47 


40 


60 


80 


99 


18 


37 


55 


73 


13 


48 


40 


60 


8C 


99 


18 


37 


55 


73 


12 


49 


9-9999440 


-9999461 


-999948C 


-9999500 


-9999519 


•9999537 


•9999556 


9^9999573 


11 


50 


41 


61 


81 


00 


19 


3e 


56 


74 


10 


51 


41 


61 


81 


00 


19 


3E 


56 


74 


9 


52 


41 


62 


81 


01 


20 


3E 


56 


74 


8 


53 


42 


62 


85 


01 


20 


3e 


57 


7£ 


7 


54 


42 


62 


85 


01 


2C 


3£ 


57 


7£ 


6 


55 


42 


63 


85 


02 


21 


3c 


57 


7£ 


5 


56 


43 


63 


Bl 


1 02 


21 


3c 


5e 


7£ 


4 


57 


43 


63 


8: 


} 02 


21 


4C 


5E 


76 


3 


58 


43 


64 


q: 


{ 03 


21 


4C 


5£ 


76 


2 


59 


9-9999444 


•9999464 


•999948^ 


I -9999503 


•9999525 


, -9999540 


) •999955c 


9^9999576 


I 


60 


44 


64 


84 03 


25 


41 


5£ 


77 





// 


55' 


54' 


53' 52' 


51' 


50' 


49' 


48' 


// 


1 LOG. COSINE 0°. 1 



Table ii.] log. tan. 89°. 147]| 


// 


4' 


5' 


6' 


7' 


8' 


9' 


10' 


11' 


"77 





11-788047^ 


7958741 


8038444 


8119636 


8202374 


8286718 


8372733 


11-8460484 


60 


1 


176^ 


796005E 


9785 


8121001 


^ 3767 


Ah 8138 


^ 4181 


1962 


59 


2 


3061 


^ 1375 


8041126 


^ 2369 


■" 5159 


^ 9558 


-^ 562g 


3440 


58 


3 


435^ 


-" 2692 


^ 2468 


^ 3736 


6553 


8290979 


7076 


4918 


57 


4 


564C 


401C 


■^ 3810 


5103 


7946 


2400 


8526 


6397 


56 


5 


694c 


532g 


5152 


6471 


9341 


3822 


9976 


7877 


55 


6 


823£ 


6646 


6495 


7839 


8210735 


5244 


8381426 


9357 


54 


7 


9532 


7965 


7839 


9208 


2130 


6666 


287C 


11-8470838 


53 


8 


ll-789082£ 


9284 


9182 


8130577 


3526 


808S 


4331 


2319 


52 


9 


212£ 


7970604 


8050527 


1947 


4922 


951S 


578S 


3801 


51 


10 


3421 


1924 


1871 


3317 


6318 


8300936 


723E 


5283 


50 


11 


471E 


3244 


3216 


4687 


7715 


2361 


8686 


6766 


49 


12 


601£ 


4565 


4561 


6058 


9113 


3786 


8390145 


8249 


48 


13 


7312 


5887 


5907 


7429 


8220510 


5211 


1596 


9733 


47 


14 


8611 


7208 


7254 


8801 


1909 


6637 


305C 


11-8481217 


46 


15 


990S 


8530 


8600 


8140173 


3307 


8063 


450£ 


2701 


45 


16 


ll-790120e 


9853 


9947 


1546 


4706 


9496 


5966 


4187 


44 


17 


2507 


7981176 


8061295 


2919 


6106 


8310917 


7416 


5672 


43 


18 


3807 


2499 


2643 


4292 


7506 


2344 


8875 


3159 


42 


19 


5107 


3823 


3991 


5666 


8906 


3772 


840032C 


8645 


41 


20 


6407 


5147 


5340 


7041 


8230307 


5201 


1787 


11-8490133 


40 


21 


7708 


6471 


6689 


8415 


1709 


6636 


324^ 


1620 


39 


22 


9009 


7796 


8039 


9791 


3111 


806C 


470S 


3109 


38 


23 


11-7910310 


9121 


9389 


8151166 


4513 


9490 


6161 


4598 


37 


24 


1612 


7990447 


8070739 


2542 


5916 


8320920 


7621 


6087 


36 


25 


2914 


1773 


2090 


3919 


7319 


2351 


9086 


7577 


35 


26 


4217 


3100 


3441 


5296 


8722 


' 3782 


8410541 


9067 


34 


27 


5520 


4427 


4793 


6673 


8240127 


5214 


2001 


11-8500558 


33 


28 


6824 


5754 


6145 


8051 


1531 


6647 


3462 


2049 


32 


29 


8127 


7082 


7497 


9429 


2936 


8079 


4924 


3541 


31 


30 


9432 


8410 


8850 


8160808 


4342 


9513 


6387 


5033 


30 


31 


11-7920736 


9738 


8080204 


2187 


5748 


8330946 


7849 


6526 


29 


32 


2041 


8001067 


1558 


3567 


7154 


2381 


9313 


8020 


28 


33 


3347 


2397 


2912 


4947 


8561 


3815 


8420776 


9513 


27 


34 


4652 


3727 


4266 


6327 


9968 


5251 


2241 


11-8511008 


26 


35 


5959 


5057 


5621 


7708 


8251376 


6686 


3705 


2503 


25 


36 


7265 


6387 


6977 


9090 


2784 


8122 


5170 


3998 


24 


37 


8572 


7718 


8333 


8170471 


4193 


9559 


6636 


5494 


23 


38 


9880 


9050 


9689 


1854 


5602 


8340996 


8102 


6991 


22 


39 


1I-7931187 


8010381 


8091046 


3236 


7011 


2434 


9569 


8488 


21 


40 


2495 


1714 


2403 


4619 


8421 


3872 


8431036 


9985 


20 


41 


3804 


: 3046 


3761 


6003 


9832 


5310 


2504 


11-8521483 


19 


42 


5113 


4379 


5119 


7387 


8261243 


6749 


3972 


2982 


18 


43 


6422 


; 5713 


6477 


8771 


2654 


8188 


5441 


4481 


17 


44 


7732 


7047 


7836 


8180156 


4066 


9628 


6910 


5980 


16 


45 


9042 


8381 


9195 


1541 


5478 


8351069 


8379 


7480 


15 


46 


11-7940353 


9716 


8100555 


2927 


6891 


2510 


9849 


8981 


14 


47 


1663 


8021051 


1915 


4313 


8304 


3951 


8441320 


U-8530482 


13 


48 


2975 


2386 


3276 


5700 


9718 


5393 


2791 


1984 


12 


49 


4286 


3722 


4637 


7087 


8271132 


6835 


4263 


3486 


11 


60 


5599 


5058 


5998 


8475 


2547 


8278 


5735 


4989 


10 


51 


6911 


6395 


7360 


9863 


3962 


9721 


7208 


6492 


9 


52 


8224 


7732 


8722 


8191251 


5377 


8361165 


8681 


7996 


8 


53 


9537 


9070 


8110085 


2640 


6793 


2609 


8450154 


9500 


7 


54 


11-7950851 


80o0408 


1448 


4029 


8210 


4054 


1629 


11-8541005 


6 


55 


2165 


1746 


2812 


5419 


9627 


5499 


3103 


2510 


5 


56 


3479 


3085 


4176 


6809 


8281044 


6945 


4578 


4016 


4 


57 


4794 


4424 


5540 


8200 


2462 


8391 


6054 


5522 


3 


58 


6110 


5764 


6905 


9591 


3880 


9838 


7530 


7029 


2 


59 


7425 


7104 


8270 


3200982 


5299U 


3371285 


9007 


8536 


1 


; 60 


8741 


8444 


9636 


2374 


6718 


2733 


8460484 


11-8550044 





' // 


55' 


54' 


53' 


52' 51' 1 50' 1 


49' 48' 1 


// 


sss 


LOG. COTAN. 0°. II 



148 LOG. SINE 89°. [Tod/e 11. Il 


" 


1 12' 


1 13' 


1 14' 


15' 


1 16' 


17' 


1 18' 1 19' 


// 





9-999957' 


1 -9999594-999961 


L -9999628 -9999644 


-9999660l-9999676l9-999969] 


60 


1 


r 


1 ^ 94 OS 1 


o. 28 °* 4f 


>=^. 6] 


P 76 9] 


59 


2 


77| 95| 12| 281 4^ 


> 6] 


76 9S 


58 


3 


7^ 


J 95 12 29 4J 


6] 


r 


' 92 


57 


4 


7f 


\ 95 12 29 4£ 


61 


r 


' 92 


56 


5 


7E 


\ 96 1- 


} 291 46 


6S 


Ti 


92 


55 


6 


7e 


96 12 


3( 


) 4£ 


65 


Ti 


93 


54 


7 


7C 


9( 


) Ic 


3( 


) 46 


6S 


7E 


\ 93 


53 


8 


7c 


96 


Ic 


3( 


) 46 


6S 


7E 


\ 93 


52 


9 


9'9999579 


-9999597 


■9999614 


-999963C 


) -9999647 


•9999662 


•9999678l9^9999693 


51 


10 


80 


97 


14 


31 


47 


63 


7£ 


94 


50 


11 


80 


97 


14 


31 


47 


63 


7£ 


94 


49 


12 


80 


9S 


16 


31 


47 


63 


7£ 


94 


48 


13 


80 


98 


15 


31 


48 


64 


7£ 


94 


47 


14 


81 


98 


15 


32 


48 


64 


7£ 


95 


46 


15 


81 


98 


15 


32 


48 


64 


8C 


95 


45 


16 


81 


99 


16 


32 


49 


64 


80 


95 


44 


17 


82 


99 


16 


33 


49 


65 


80 


95 


43 


18 


82 


99 


16 


33 


49 


65 


80 


96 


42 


19 


9-9999582 


-9999600 


•9999617 


■9999633 


-9999649 


•9999665 


•9999681 


9-9999696 


41 


20 


83 


00 


17 


33 


50 


66 


81 


96 


40 


21 


83 


00 


17 


34 


50 


66 


81 


^^ 


39 


22 


83 


00 


17 


34 


50 


66 


82 


97 


38 


23 


83 


01 


18 


34 


50 


66 


82 


97 


37 


24 


84 


01 


18 


34 


51 


67 


82 


97 


36 


25 


84 


01 


18 


35 


51 


67 


82 


97 


35 


26 


84 


02 


18 


35 


51 


67 


83 


98 


34 


27 


85 


02 


19 


35 


52 


67 


83 


98 


33 


28 


85 


02 


19 


36 


52 


68 


83 


98 


32 


29 


9-9999585 


•9999602 


•9999619 


•9999636 


•QOQOCEO 


•9999668 


•9999683 


9-9999698 


31 


30 


85 


03 


20 


36 


52 


68 


84 


99 


30 


31 


86 


03 


20 


36 


53 


68 


84 


99 


29 


32 


86 


03 


20 


37 


53 


69 


84 


99 


28 


33 


86 


04 


20 


37 


53 


69 


84 


99 


27 


34 


87 


04 


21 


37 


53 


69 


85 


9-9999700 


26 


35 


87 


04 


21 


37 


54 


69 


85 


00 


25 


36 


87 


04 


21 


38 


54 


70 


85 


00 


24 


37 


87 


05 


22 


38 


54 


70 


85 


00 


23 


38 


88 


05 


22 


38 


54 


70 


86 


01 


22 


39 


9-9999588 


-9999605 


•9999622 


•9999639 


•9999655 


•9999670 


•9999686 


9-9999701 


21 


40 


88 


06 


22 


39 


55 


71 


86 


01 


20 


41 


89 


06 


23 


39 


55 


71 


86 


01 


19 


42 


89 


06 


23 


39 


55 


71 


87 


02 


18 


43 


89 


06 


23 


40 


56 


71 


87 


02 


17 


44 


89 


07 


23 


40 


56 


72 


87 


02 


16 


45 


90 


07 


24 


40 


56 


72 


87 


02 


15 


46 


90 


07 


24 


40 


57 


72 


88 


03 


14 


47 


90 


08 


24 


41 


57 


73 


88 


03 


13 


48 


91 


08 


25 


41 


57 


73 


88 


03 


12 


49 


9-9999591 


9999608 


9999625 


-9999641 


•9999657 


9999673 


9999688 


9-9999703 


11 


50 


91 


08 


25 


42 


58 


73 


89 


04 


10 


51 


92 


09 


25 


42 


58 


74 


89 


04 


9 


52 


92 


09 


26 


42 


58 


74 


89 


04 


8 


53 


92 


09 


26 


42 


58 


74 


89 


04 


7 


54 


92 


09 


26 


43 


59 


74 


90 


05 


6 


55 


93 


10 


27 


43 


59 


75 


90 


05 


5 


56 


93 


10 


27 


43 


59 


75 


90 


05 


4 


57 


93 


10 


27 


44 


59 


75 


90 


05 


3 


58 


94 


11 


27 


44 


60 


75 


91 


06 


2 


59 


9'9999594 


9999611 


9999628 


9999644 


9999660 • 


9999676 


9999691 ( 


)-9999706 11 


60 


94 


11 


28 


44 


60 76| 


91 


06 Ol 


// 


4r 


46' 45' I 


44' 


43' 42' 1 41' 1 40' I'M! 


LOG. COSINE 0°. V 



Table 11.] LOG. TAN. 89°. 149] 




// 


12' 


13' 


14' 


15' 


16' 


17' 


18' 


19' 


" 







11-8550044 


8641490 


8734901 


8830366 


8927975 


9027828 


9130030 


11-9234694 


60 




1 


1553 


^ 3030 


^ 6475 


^ 1975 


9620 


9512 


- 1754 


6460 


59 




2 


3062 


'^ 4571 


'^ 8050 


-" 3584 


8931267 


9031196 


^ 3478 


8227 


58 




3 


4571 


6113 


9625 


5194 


- 2913 


-^ 2881 


5204 


9994 


57 




4 


6081 


7655 


8741201 


6805 


^ 4561 


^ 4567 


6930 


11-9241762 


56 




5 


7592 


9198 


2777 


8416 


6209 


6253 


8656 


3531 


55 




6 


9103 


8650741 


4354 


8840028 


7858 


7940 


9140384 


5301 


54 




7 


11-8560614 


2285 


5931 


1641 


9507 


9628 


2112 


7071 


53 




8 


2126 


3829 


7509 


3254 


8941157 


9041317 


3840 


8842 


52 




9 


3639 


5374 


9088 


4868 


2807 


3006 


5570 


11-9250614 


51 




10 


5152 


6919 


8750667 


6482 


4458 


4695 


7300 


2386 


50 




11 


6666 


8465 


2247 


8097 


6110 


6386 


9031 


4159 


49 




12 


8180 


8660012 


3827 


9713 


7763 


8077 


9150762 


5933 


48 




13 


9695 


1559 


5408 


8851329 


9416 


9768 


2494 


7708 


47 




14 


11-8571210 


3107 


6989 


2946 


8951069 


9051461 


4227 


9483 


46 




15 


2726 


4655 


8571 


4563 


2724 


3154 


5961 


ir9261259 


45 




16 


4242 


6204 


8760154 


6181 


4379 


4847 


7695 


3036 


44 




17 


5759 


7753 


1737 


7800 


6034 


6542 


9430 


4814 


43 




18 


7276 


9303 


3321 


9419 


7691 


8237 


9161165 


6592 


42 




19 


8794 


8670853 


4905 


8861039 


9347 


9932 


2902 


8371 


41 




20 


11-8580313 


2404 


6490 


2659 


8961005 


9061629 


4639 


11-9270150 


40 




21 


1832 


3956 


8076 


4280 


2663 


3326 


6376 


1931 


39 




22 


3351 


5508 


9662 


5902 


4322 


5023 


8115 


3712 


38 




23 


4871 


7060 


8771248 


7524 


5981 


6722 


9854 


5494 


37 




24 


6392 


8614 


2836 


9147 


7641 


8421 


8171594 


7277 


36 




25 


7913 


8680167 


4423 


8870770 


9302 


9070120 


3334 


9060 


35 




26 


9434 


1722 


6012 


2394 


8970963 


1820 


5075 


11-9280844 


34 




27 


11-8590957 


3277 


7601 


4019 


2625 


3521 


6817 


2629 


33 




28 


2479 


4832 


9190 


5644 


4287 


5223 


8560 


4414 


32 




29 


4003 


6388 


8780781 


7270 


5951 


6925 


9180303 


6201 


31 




30 


5526 


7944 


2371 


8896 


7614 


8628 


2047 


7988 


30 




31 


7051 


9502 


3963 


8880523 


9279 


9080332 


3792 


9775 


29 




32 


8575 


8691059 


5554 


2151 


8980944 


2036 


5537 


11-9291564 


28 




33 


11-8600101 


2617 


7147 


3779 


2610 


3741 


7283 


3353 


27 




34 


1627 


4176 


8740 


5408 


4276 


5447 


9030 


5143 


26 




35 


3153 


5735 


8790334 


7038 


5943 


7153 


9190777 


6934 


25 




36 


4680 


7295 


1928 


8668 


7611 


8860 


2525 


8725 


24 




37 


6208 


8856 


3523 


8890298 


9279 


9090568 


4274 


11-9300517 


23 




38 


7736 


8700417 


5118 


1930 


8990948 


2276 


6024 


2310 


22 




39 


9264 


1978 


6714 


3562 


2618 


3985 


7774 


4104 


21 




40 


11-8610793 


3540 


8311 


5194 


4288 


5695 


9525 


5898 


20 




41 


2323 


5103 


9908 


6827 


5959 


7405 


9201277 


7694 


19 




42 


3853 


6666 


8801505 


8461 


7630 


9116 


3029 


9489 


18 




43 


5384 


8230 


3104 


8900096 


9302 


9100828 


4782 


11-9311286 


17 




44 


6915 


9794 


4703 


1731 


9000975 


2540 


6536 


3083 


16 




45 


8447 


8711359 


6302 


3366 


2649 


4253 


8291 


4882 


15 




46 


9980 


2925 


7902 


5002 


4323 


5967 


9210046 


6680 


14 




47 


11-8621512 


4491 


9503 


6639 


5997 


7681 


1802 


8480 


13 




48 


3046 


6058 


8811104 


8277 


7673 


9396 


3559 


11-9320280 


12 




49 


4580 


7625 


2706 


9915 


9349 


9111112 


5316 


2081 


11 




50 


6114 


9193 


4309 


8911554 


9011025 


2828 


7074 


3883 


10 




51 


7650 


8720761 


5912 


3193 


2703 


4545 


8833 


5686 


9 




52 


9185 


2330 


7515 


4833 


4381 


6263 


9220593 


7489 


8 




53 


11-8630721 


3899 


9119 


6474 


6059 


7982 


2353 


9293 


7 




54 


2258 


5469 


8820724 


8115 


7739 


9701 


4114 


11-9331098 


6 




55 


3795 


7040 


?330 


9757 


9418 


9121421 


5875 


2904 


5 




56 


5333 


8611 


3936 


8921399 


9021099 


3141 


7638 


4710 


4 




57 


6871 


8730183 


5542 


3042 


2780 


4862 


9401 


6517 


3 




58 


8410 


1755 


7149 


4686 


4462 


6584 


9231165 


8325 


2 




59 


9950 


3328 


8757 


6330 


6145 


8307 


2929 


11-9340134 


1 




60 


11-8641490 


4901 


8830366 


7975 


7828 9130030 


4694 


1943 







" 


47' 


46' 


45' 


44' 


43' 42' 


41' 


40' 


" 







LOG. COTAN. 0°. 





13* 



150 LOG. SINE890. [Table n. j 




// 


20' 


21' 


22' 


23' 


24' 


25' 


26' 


27' 


// 







9-9999706 


•9999721 


•9999735 


•9999748 


•9999762 


-9999775 


•9999788 


9^9999800 


60 




1 


06 


^ 21 


^ 35 


^ 49 


=^ 62 


^ 75 


<^ 88 


00 


59 




2 


06 


21 


35 


49 


62 


75 


88 


00 


58 




3 


07 


21 


35 


49 


63 


76 


88 


01 


57 




4 


07 


21 


36 


49 


63 


76 


88 


01 


56 




5 


07 


22 


36 


50 


63 


76 


89 


01 


55 




6 


07 


22 


36 


50 


63 


76 


89 


01 


54 




7 


08 


22 


36 


50 


. 63 


76 


89 


01 


53 




8 


08 


22 


37 


50 


64 


77 


89 


02 


52 




9 


08 


23 


37 


50 


64 


77 


89 


02 


51 




10 


9-9999708 


•9999723 


-9999737 


•9999751 


•9999764 


-9999777 


•9999790 


9-9999802 


50 




11 


09 


23 


37 


51 


64 


77 


90 


02 


49 




12 


09 


23 


37 


51 


65 


77 


90 


02 


48 




13 


09 


24 


38 


51 


65 


78 


90 


03 


47 




14 


09 


24 


38 


52 


65 


78 


90 


03 


46 




15 


10 


24 


38 


52 


65 


78 


91 


03 


45 




16 


10 


24 


38 


52 


65 


78 


91 


03 


44 




17 


10 


25 


39 


52 


66 


79 


91 


03 


43 




18 


10 


25 


39 


53 


66 


79 


91 


04 


42 




19 


11 


25 


39 


53 


66 


79 


92 


04 


41 




20 


9-9999711 


•9999725 


•9999739 


•9999753 


•9999766 


-9999779 


■9999792 


9-9999804 


40 




21 


11 


26 


40 


53 


66 


79 


92 


04 


39 




22 


11 


26 


40 


53 


67 


80 


92 


04 


38 




23 


12 


26 


40 


54 


67 


80 


92 


05 


37 




24 


12 


26 


40 


54 


67 


80 


93 


05 


36 




25 


12 


26 


40 


54 


67 


80 


93 


05 


35 




26 


12 


27 


41 


54 


68 


80 


93 


05 


34 




27 


13 


27 


41 


55 


68 


81 


93 


05 


33 




28 


13 


27 


41 


55 


68 


81 


93 


06 


32 




29 


13 


27 


41 


55 


68 


81 


94 


06 


31 




30 


9-9999713 


-9999728 


•9999742 


•9999755 


•9999768 


•9999781 


•9999794 


9-9999806 


30 




31 


14 


28 


42 


55 


69 


82 


94 


06 


29 




32 


14 


28 


42 


56 


69 


82 


94 


06 


28 




33 


14 


28 


42 


56 


69 


82 


94 


07 


27 




34 


14 


29 


43 


56 


69 


82 


95 


07 


26 




35 


15 


29 


43 


56 


70 


82 


95 


07 


25 




36 


15 


29 


43 


57 


70 


83 


95 


07 


24 




37 


15 


29 


43 


57 


70 


83 


95 


07 


23 




38 


15 


30 


43 


57 


70 


83 


95 


06 


22 




39 


15 


30 


44 


57 


70 


83 


96 


0£ 


21 




40 


9-9999716 


-9999730 


-9999744 


-9999757 


-9999771 


•9999783 


•9999796 


9-999980G 


20 




41 


16 


30 


44 


58 


71 


84 


96 


OS 


19 




42 


16 


30 


44 


58 


71 


84 


96 


oe 


18 




43 


16 


31 


45 


58 


71 


84 


96 


oe 


17 




44 


17 


31 


45 


58 


71 


84 


97 


OS 


16 




45 


17 


31 


45 


59 


72 


84 


97 


OS 


15 




46 


17 


31 


45 


59 


72 


85 


97 


09 


14 




47 


17 


32 


45 


59 


72 


85 


97 


09 


13 




48 


18 


32 


46 


59 


72 


85 


97 


09 


12 




49 


18 


32 


46 


59 


73 


85 


98 


10 


11 




50 


9-9999718 


-9999732 


-9999746 


-9999760 


-9999773 


•9999786 


•9999798 


9-9999810 


10 




51 


18 


33 


46 


60 


73 


86 


98 


10 


9 




52 


19 


33 


47 


60 


73 


86 


98 


10 


8 




53 


19 


33 


47 


60 


73 


86 


98 


10 


7 




54 


19 


33 


47 


61 


74 


86 


99 


11 


6 




55 


19 


34 


47 


61 


74 


87 


99 


11 


5 




56 


20 


34 


48 


61 


74 


87 


99 


11 


4 




57 


20 


34 


48 


61 


74 


87 


99 


11 


3 




58 


20 


34 


48 


61 


74 


87 


99 


11 


2 




59 


20 


34 


48 


62 


75 


87 


•9999800 


12 


1 




60 


21 


35 


48 


62 


75 


88 


00 


12 







'' 


39' 


38' 


37' 


36' 


35' 


34' 


33' 


32' 


// 




LOG. COSINE 0°. i 





Table ii.] log. tan. 89°. 151 {| 


// 


2<y 1 21' 


22' 


23' 


24' 


25' 


26' 27' 1 


'' 





11-9341943 


9451906 


9564726 


9680554 


9799555 


11-9921908 


12-0047808 


12-0177466 


60 


1 


3753 


^ 3763 


^ 6631 


^ 2511 


9801566 


3977 


9938 


9660 


59 


2 


5564 


'" 5620 


^ 8538 


^ 4469 


^ 3578 


6047 


12-0052068 


12-0181855 


58 


3 


7375 


7478 


9570445 


6427 


" 5592 


8117 


4200 


4052 


57 


4 


9188 


9337 


2352 


8387 


7606 


11-9930189 


6333 


6249 


56 


5 


11-9351001 


9461197 


4261 


9690347 


9621 


2262 


8466 


8448 


55 


6 


2815 


3057 


6171 


2308 


9811636 


4335 


12-0060601 


12-0190647 


54 


7 


4629 


4919 


8081 


4271 


3653 


6410 


2737 


2848 


53 


8 


6445 


6781 


9992 


6234 


5671 


8486 


4874 


5050 


52 


9 


8261 


8644 


9581904 


8198 


7690 


11-9940562 


7012 


7253 


51 


10 


11-9360078 


9470507 


3817 


9700162 


9709 


2640 


9151 


9467 


50 


11 


1896 


2372 


5731 


2128 


9821730 


4718 


12-0071291 


12-0201662 


49 


12 


3714 


4237 


7646 


4095 


3752 


6768 


3432 


3869 


48 


13 


5533 


6103 


9561 


6062 


5774 


8879 


5574 


6076 


47 


14 


7353 


7970 


9591478 


8031 


7797 


11-9950960 


7717 


8285 


46 


15 


9174 


9838 


3395 


9710000 


9822 


3043 


9862 


12-0210494 


45 


16 


11-9370995 


9481706 


5313 


1970 


9831847 


5126 


12-0082007 


2705 


44 


17 


2818 


3576 


7232 


3941 


3873 


7211 


4153 


4917 


43 


18 


4641 


5446 


9151 


5913 


5901 


9297 


6301 


7130 


42 


19 


6464 


7317 


9601072 


7886 


7929 


11-9961383 


8449 


9345 


41 


20 


8289 


9188 


2993 


9860 


9958 


3471 


12-0090599 


12-0221560 


40 


21 


11-9380114 


9491061 


4915 


9721834 


9841988 


5559 


2749 


3776 


39 


22 


1940 


2934 


6838 


3810 


4019 


7649 


4901 


5994 


38 


23 


3767 


4808 


8762 


5787 


6051 


9740 


7054 


8213 


37 


24 


5595 


6683 


9610687 


7764 


8084 


11-9971831 


9208 


12-0230433 


36 


25 


7423 


8559 


2613 


9742 


9850117 


3924 


12-0101363 


2654 


35 


26 


9252 


9500436 


4539 


9731721 


2152 


6017 


3519 


4876 


34 


27 


11-9391082 


2313 


6467 


3701 


4188 


8112 


5676 


7099 


33 


28 


2913 


4191 


8395 


5682 


6225 


11-9980208 


7834 


9324 


32 


29 


4745 


6070 


9620324 


7664 


8262 


2304 


9993 


12-0241549 


31 


30 


6577 


7950 


2254 


9647 


9860301 


4402 


12-0112153 


3776 


30 


31 


8410 


9831 


4185 


9741631 


2340 


6501 


4315 


6004 


29 


32 


11-9400244 


9511712 


6116 


3615 


4381 


8600 


6477 


8233 


28 


33 


2078 


3594 


8049 


5601 


6422 


11-9990701 


8641 


12-0250463 


27 


34 


3913 


5477 


9982 


7587 


8465 


2803 


12-0120805 


2694 


26 


:35 


5750 


7361 


9631916 


9574 


9870508 


4906 


2971 


4927 


25 


i36 


7586 


9246 


3851 


9751563 


2553 


7009 


5138 


7160 


24 


37 


9424 


9521131 


5787 


3552 


4598 


9114 


7305 


9395 


23 


38 


11-9411263 


3018 


7724 


5542 


6644 


12-0001220 


9474 


12-0261631 


22 


'39 


3102 


4905 


9662 


7533 


8692 


3327 


12-0131644 


3868 


21 


40 


4942 


6793 


9641600 


9525 


9880740 


5435 


3815 


6106 


20 


41 


6783 


8682 


3540 


9761517 


2789 


7544 


5987 


8345 


19 


42 


8624 


9530571 


5480 


3511 


4839 


9654 


8161 


12-0270586 


18 


43 


11-9420466 


2462 


7421 


5506 


6890 


12-0011764 


12-0140335 


2827 


17 


44 


2310 


4353 


9363 


7501 


8942 


3876 


2510 


5070 


16 


45 


4154 


6245 


9651306 


9498 


9890995 


5989 


4687 


7314 


15 


46 


5998 


8138 


3250 


9771495 


3049 


8103 


6865 


9559 


14 


47 


7844 


9540032 


5194 


3493 


5104 


12-0020218 


9043 


12-0281806 


13 


48 


9690 


1926 


7140 


5493 


7160 


2334 


12-0151223 


4053 


12 


49 


11-9431537 


3822 


9086 


7493 


9217 


4452 


3404 


6302 


11 


50 


3385 


5718 


9661033 


9494 


9901275 


6570 


5586 


8551 


10 


51 


5233 


7615 


2982 


9781496 


3334 


8689 


7769 


12-0290802 


9 


52 


7083 


9513 


4931 


3499 


5394 


12-0030809 


9953 


3055 


8 


53 


8933 


9551412 


6880 


5502 


7455 


2930 


120162138 


5308 


7 


54 


11-9440784 


3311 


8831 


7507 


9517 


5053 


4325 


7562 


6 


55 


2636 


5212 


9670783 


9513 


9911580 


7176 


6512 


9818 


5 


56 


4488 


7113 


2735 


9791519 


3643 


9300 


8701 


12-0302075 


4 


57 


6342 


9015 


4689 


3527 


5708 


12-0041426 


12.0170890 


4333 


3 


58 


8196 


9560918 


6643 


5535 


7774 


3552 


3081 


6592 


2 


59 


11-9450051 


2821 


8598 


7545 


9841 


5680 


5273 


8852 


1 


60 


1906 


4726 


9680554 


9555 


9921908 


7808 


7466 


12-0311114 





" 


39' 


38' 


37' 


36' 


35' 


34' 


33' 


32' 


" 


LOG. COTAN. Oo. || 



152 LOG. SINE 89°. [Table 


n. 


// 


28' 


29' 


30' 


31' 


32' 


33' 


34' 


35' 


"/" 





9-9999812 


•9999823 


-9999835 


9999845 


•9999856 


-9999866 


9999876 


9-9999885 


60 


1 


12 


oi 24 


^ 35 


^ 46 


^ 56 


^ 66 


^ 76 


85 


59 


2 


12 


24 


35 


46 


56 


66 


76 


85 


58 


3 


12 


24 


35 


46 


56 


67 


76 


86 


57 


4 


13 


24 


35 


46 


57 


67 


76 


86 


56 


5 


13 


24 


36 


46 


57 


67 


77 


86 


55 


6 


13 


25 


36 


47 


57 


67 


77 


86 


54 


7 


13 


25 


36 


47 


57 


67 


77 


86 


53 


8 
9 


13 


25 


36 


47 


57 


67 


77 


86 


52 


9-9999814 


9999825 


•9999836 


9999847 


•9999857 


-9999868 


■9999877 


9-9999887 


51 


10 
11 
12 


14 


25 


36 


47 


58 


68 


77 


87 


50 


14 


26 


37 


47 


58 


68 


78 


87 


49 


14 


26 


37 


48 


58 


68 


78 


87 


48 


13 
14 


14 


26 


37 


48 


58 


68 


78 


87 


47 


15 


26 


37 


48 


58 


68 


78 


87 


46 


15 


15 


26 


37 


48 


59 


69 


78 


87 


45 


16 


15 


26 


38 


48 


59 


69 


78 


88 


44 


17 


15 


27 


38 


48 


59 


69 


78 


88 


43 


18 


15 


27 


38 


49 


59 


69 


79 


88 


42 


19 


9-9999816 


9999827 


-9999838 


9999849 


•9999859 


-9999869 


•9999879 


9-9999a88 


41 


20 


16 


27 


38 


49 


59 


69 


79 


88 


40 


21 


16 


27 


38 


49 


60 


70 


79 


88 


39 


22 


16 


28 


39 


49 


60 


70 


79 


89 


38 


23 


16 


28 


39 


50 


60 


70 


79 


89 


37 


24 


17 


28 


39 


50 


60 


70 


80 


89 


36 


25 


17 


28 


39 


50 


60 


70 


80 


89 


35 


26 


17 


28 


39 


50 


60 


70 


5-0 


89 


34 


27 


17 


29 


40 


50 


61 


70 


80 


89 


33 


28 


17 


29 


40 


50 


61 


71 


80 


89 


32 


29 


9-9999817 


-9999829 


•9999840 


■9999851 


-9999861 


•9999871 


-9999880 


9-9999890 


31 


30 


18 


29 


40 


51 


61 


71 


81 


90 


30 


31 


18 


29 


40 


51 


61 


71 


81 


90 


29 


32 


18 


29 


40 


51 


61 


71 


81 


90 


28 


33 


18 


30 


41 


51 


62 


71 


81 


90 


27 


34 


18 


30 


41 


51 


62 


72 


81 


90 


26 


35 


19 


30 


41 


52 


62 


72 


81 


90 


25 


36 


19 


30 


41 


52 


62 


72 


81 


91 


24 


37 


19 


30 


41 


52 


62 


72 


82 


91 


23 


38 


19 


31 


42 


52 


62 


72 


82 


91 


22 


39 


9-9999819 


-9999831 


-9999842 


■9999852 


•9999863 


•9999872 


•9999882 


9-9999891 


21 


40 


20 


31 


42 


52 


63 


73 


82 


91 


20 


41 


20 


31 


42 


53 


63 


73 


82 


91 


19 


42 


20 


31 


42 


53 


63 


73 


82 


92 


18 


43 


20 


31 


42 


53 


63 


73 


83 


92 


17 


44 


20 


32 


43 


53 


63 


73 


83 


92 


16 


45 


21 


32 


43 


53 


64 


73 


83 


92 


15 


46 


21 


32 


43 


54 


64 


74 


83 


92 


14 


47 


21 


32 


43 


54 


64 


74 


83 


92 


13 


48 


21 


32 


43 


54 


64 


74 


83 


92 


12 


49 


9-9999821 


•9999833 


•9999844 


•9999854 


•9999864 


•9999874 


•9999883 


9-9999893 


11 


50 


22 


33 


44 


54 


64 


74 


84 


93 


10 


51 


22 


33 


44 


54 


65 


74 


84 


93 


9 


52 


22 


33 


44 


55 


65 


75 


84 


93 


8 


53 


22 


33 


44 


55 


65 


75 


84 


&3 


7 


54 


22 


34 


44 


55 


65 


75 


84 


93 


6 


55 


22 


34 


45 


55 


65 


75 


84 


93 


5 


56 


23 


34 


45 


55 


65 


75 


85 


94 


4 


57 


23 


34 


45 


55 


66 


75 


85 


94 


3 


58 


23 


34 


45 


56 


66 


75 


85 


94 


2 


59 


9-9999823 


-9999834 


-9999845 


•9999856 


-9999866 


•9999876 


•9999885 


9-9999894 


I 


60 


23 


35 


45 


56 


66 


76 


85 


94 





'/ 


31' 


30' 


29' 


28' 


27' 


26' 


25' 


24' 


" 


1 LOG. COSINE 0°. || 



^able II.] LOG. TAN. 89°. 






153 j 


// 


28' 


29' 


30' 


31' 


32' 


33' 


34' 


35' 


// 





12-0311114 


0449004 


0591416 


0738656 


0891062 


1049012 


1212923 


12-1383262 


60 


1 


3376 


0451340 


^ 3830 


0741153 


^93646 


<^ 51694 


(^ 15707 


86159 


59 


2 


5640 


(f^ 3677 


■^ 6244 


^ 3651 


^96236 


^54377 


^ 18494 


89057 


58 


3 


7905 


^ 6015 


8661 


^ 6150 


98825 


57062 


21283 


91957 


57 


4 


12-0320171 


8354 


0601078 


8652 


0901416 


59749 


24073 


94859 


56 


5 


2439 


0460695 


3497 


0751154 


04008 


62437 


26865 


97763 


55 


6 


4707 


3037 


5917 


3658 


06601 


65127 


29659 


12-1400669 


54 


7 


6977 


5380 


8339 


6164 


09197 


67819 


32455 


03577 


53 


8 


9248 


7725 


0610762 


8670 


1,1793 


70513 


35252 


06487 


52 


9 


12-0331520 


0470071 


3186 


0761179 


14392 


73208 


38051 


09399 


51 


10 


3794 


2418 


5612 


3688 


16992 


75904 


40853 


12313 


50 


11 


6068 


4766 


8039 


6200 


19593 


78603 


43656 


15229 


49 


12 


8344 


7115 


0620467 


8712 


22196 


81303 


46460 


18147 


48 


13 


12-0340621 


9466 


2897 


0771226 


24801 


84005 


49267 


21066 


47 


14 


2899 


0481818 


5328 


3742 


27407 


86708 


52075 


23988 


46 


15 


5178 


4172 


7761 


6259 


30015 


89413 


54885 


26912 


45 


16 


7459 


6526 


0630195 


8778 


32624 


92120 


57697 


29837 


44 


17 


9740 


8882 


2630 


0781298 


35235 


94829 


60511 


32765 


43 


18 


12-0352023 


0491240 


5066 


3819 


37847 


97539 


63327 


35695 


42 


19 


4307 


3598 


7504 


6342 


40461 


1100251 


66144 


38626 


41 


20 


6592 


5958 


9943 


8866 


43077 


02964 


68963 


41560 


40 


21 


8879 


8319 


0642384 


0791392 


45694 


05680 


71785 


44495 


39 


22 


12-0361167 


0500681 


4826 


3919 


48313 


08397 


74607 


47433 


38 


23 


3456 


3045 


7270 


6448 


50933 


11115 


77432 


50372 


37 


24 


5746 


5410 


9714 


8978 


53555 


13836 


80259 


53314 


36 


25 


8037 


7776 


0652161 


0801510 


56178 


16558 


83087 


56257 


35 


26 


12-0370330 


0510144 


4608 


4043 


58803 


19282 


85918 


59203 


34 


27 


2623 


2512 


7057 


6578 


61430 


22007 


88750 


62150 


33 


28 


4918 


4882 


9507 


9114 


64058 


24734 


91584 


65100 


32 


29 


7214 


7254 


0661959 


0811652 


66688 


27463 


94420 


68051 


31 


30 


9512 


9626 


4412 


4191 


69319 


30194 


97257 


71004 


30 


31 


12-0381810 


0522000 


6867 


6731 


71952 


32926 


1300097 


73960 


29 


32 


4110 


4376 


9322 


9273 


74587 


35661 


02938 


76917 


28 


33 


6411 


6752 


0671780 


0821817 


77223 


38396 


05782 


79877 


27 


34 


8713 


9130 


4238 


4362 


79861 


41134 


08627 


82839 


26 


35 


12-0391017 


0531509 


6698 


6909 


82500 


43873 


11474 


85802 


25 


36 


3322 


3890 


9160 


9457 


85141 


46614 


14323 


88768 


24 


37 


5627 


6272 


0681622 


0832006 


87784 


49357 


17173 


91735 


23 


38 


7935 


8655 


4087 


4557 


90428 


52101 


20026 


94705 


22 


39 


12-0400243 


0541039 


6552 


7110 


93074 


54848 


22880 


97677 


21 


40 


2553 


3425 


9019 


9664 


95721 


57596 


25737 


12-1500650 


20 


41 


4863 


5812 


0691488 


0842219 


98370 


60345 


28595 


03626 


19 


42 


7175 


8200 


3957 


4776 


1001021 


63097 


31455 


06604 


18 


43 


9489 


0550590 


6429 


7335 


03673 


65850 


34317 


09584 


17 


44 


12-0411803 


2981 


8901 


9895 


06327 


68605 


37181 


12565 


16 


45 


4119 


5373 


0701375 


0852457 


08983 


71361 


40047 


15549 


15 


46 


6436 


7767 


3851 


5020 


11640 


74120 


42915 


18535 


14 


47 


8754 


0560161 


6328 


7584 


14299 


76880 


45784 


21523 


13 


48 


12-0421074 


2558 


8806 


0860150 


16959 


79642 


48656 


24513 


12 


49 


3394 


4955 


0711286 


2718 


19621 


82406 


51529 


27506 


11 


50 


5716 


7354 


3767 


5287 


22285 


85171 


54404 


30500 


10 


51 


8039 


9754 


6249 


7858 


24950 


87938 


57281 


33496 


9 


52 


12-0430364 


0572156 


8733 


0870430 


^7617 


90707 


60161 


36494 


8 


53 


2690 


4559 


0721218 


3004 


30286 


93478 


63042 


39495 


7 


54 


5016 


6963 


3705 


55^9 


32956 


96250 


65924 


42497 


6 


55 


7345 


9368 


6193 


8156 


35628 


99025 


68809 


45502 


5 


56 


9674 


0581775 


8683 


0880734 


38301 


1201801 


71696 


48508 


4 


57 


12-0442005 


4183 


0731174 


3314 


40977 


04578 


74585 


51517 


3 


58 


4337 


6593 


3667 


5895 


43653 


07358 


77475 


54528 


2 


59 


6670 


9004 


6160 


8478 


46332 


10139 


80368 


57541 


1 


60 


9004 


0591416 


8656 


0891062 


49012 


12923 


83262 


60556 





" 


3r 


30' 


29' 


28' 


27' 


26' 


25' 


24' 


// 





LOG. CO 


TAN. 0= 






! 


hmmm. 










-^ 











154 LOG. SINE 89°. [Table 


n. 


36' 1 


37' 


38' 


39' 


40' 41' 1 


42' 


43' 1 


~" 





9-9999894 


9999903 


-9999911 


9999919 


9999927 


9999934 


•9999940 


9^9999947 


60 


1 


94 


^ 03 


o 11 


^ 19 


=* 27 


^ 34 


o 41 


47 


59 


2 


94 


03 


11 


19 


27 


34 


41 


47 


58 


3 


95 


03 


11 


19 


27 


34 


41 


47 


57 


4 


95 


03 


12 


19 


27 


34 


41 


47 


56 


5 


95 


04 


12 


20 


27 


34 


41 


47 


55 


6 


95 


04 


12 


20 


27 


34 


41 


48 


54 


7 


95 


04 


12 


20 


27 


34 


41 


48 


53 


8 


95 


04 


12 


20 


27 


35 


41 


48 


52 


9 


9-9999895 


9999904 


-9999912 


•9999920 


•9999928 


•9999935 


•9999941 


9-9999948 


51 


10 


96 


04 


12 


20 


28 


35 


42 


48 


50 


11 


96 


04 


13 


20 


28 


35 


42 


48 


49 


12 


96 


04 


13 


21 


28 


35 


42 


48 


48 


13 


96 


05 


13 


21 


28 


35 


42 


48 


47 


14 


96 


05 


13 


21 


28 


35 


42 


48 


46 


15 


96 


05 


13 


21 


28 


35 


42 


48 


45 


16 


97 


05 


13 


21 


28 


36 


42 


49 


44 


17 


97 


05 


13 


21 


29 


36 


42 


49 


43 


18 


97 


05 


13 


21 


29 


36 


43 


49 


42 


19 


9-9999897 


•9999905 


•9999914 


•9999921 


•9999929 


•9999936 


•9999943 


9-9999949 


41 


20 


97 


06 


14 


22 


29 


36 


43 


49 


40 


21 


97 


06 


14 


22 


29 


36 


43 


49 


39 


22 


97 


06 


14 


22 


29 


36 


43 


49 


38 


23 


98 


06 


14 


22 


29 


36 


43 


49 


37 


24 


98 


06 


14 


22 


29 


36 


43 


49 


36 


25 


98 


06 


14 


22 


30 


37 


43 


49 


35 


26 


98 


06 


15 


22 


30 


37 


43 


50 


34 


27 


98 


07 


15 


22 


30 


37 


43 


50 


33 


28 


98 


07 


15 


23 


30 


37 


44 


50 


32 


29 


9-9999898 


•9999907 


•9999915 


•9999923 


•9999930 


•9999937 


•9999944 


9-9999950 


31 


30 


99 


07 


15 


23 


30 


37 


44 


50 


30 


31 


99 


07 


15 


23 


30 


37 


44 


50 


29 


32 


99 


07 


15 


23 


30 


37 


44 


50 


28 


33 


99 


07 


15 


23 


30 


37 


44 


50 


27 


34 


99 


08 


16 


23 


31 


38 


44 


50 


26 


35 


99 


08 


16 


23 


31 


38 


44 


50 


25 


36 


99 


08 


16 


24 


31 


38 


44 


51 


24 


37 


9-9999900 


08 


16 


24 


31 


38 


44 


51 


23 


38 


00 


08 


16 


24 


31 


38 


45 


51 


22 


39 


9-9999900 


•9999908 


•9999916 


•9999924 


•9999931 


•9999938 


•9999945 


9-9999951 


21 


40 


00 


08 


16 


24 


31 


38 


45 


51 


20 


41 


00 


08 


17 


24 


31 


38 


45 


51 


19 


42 


00 


09 


17 


24 


32 


38 


45 


51 


18 


43 


00 


09 


17 


24 


32 


39 


45 


51 


17 


44 


01 


09 


17 


25 


32 


39 


45 


51 


16 


45 


01 


09 


17 


25 


32 


39 


45 


51 


15 


46 


01 


09 


17 


25 


32 


39 


45 


52 


14 


47 


01 


09 


17 


25 


32 


39 


46 


52 


13 


48 


01 


09 


17 


25 


32 


39 


46 


52 


12 


49 


9-9999901 


-9999910 


-9999918 


-9999925 


-9999932 


•9999939 


•9999946 


9-9999952 


11 


50 


01 


10 


18 


25 


33 


39 


46 


52 


10 


51 


02 


10 


18 


25 


33 


39 


46 


52 


9 


52 


02 


10 


18 


26 


33 


40 


46 


52 


8 


53 


02 


10 


18 


26 


33 


40 


46 


52 


7 


54 


02 


10 


18 


26 


33 


40 


46 


52 


6 


55 


02 


IC 


18 


26 


33 


40 


46 


52 


5 


56 


02 


11 


18 


26 


33 


40 


46 


53 


4 


57 


02 


11 


19 


26 


33 


40 


47 


53 


3 


58 


03 


11 


19 


26 


33 


40 


47 


53 


2 


59 


9-9999903 


-999991] 


-9999919 


•9999926 


-9999934 


•9999940 


•9999947 


9-9999953 


1 


60 


03 


1] 


IS 


27 


34 


40 


47 


53 





" 


23' 


22- 


1 21' 


20' 1 19' 


18' 


17' 


16' 


" 


LOG. COSINE 0°. 1 



Table II.] 






LOG. 


TAN. 89°. 




155 1 




" 


36' 


37' 


38' 


39' 


40' 


41' 


42' 


43' 


// 







121560556 


1745396 


1938453 


2140492 


2352390 


•2575159 


2809974 


12-3058214 


60 




1 


63573 


CM 48544 


cf.1 41744 


^43940 


(N 56011 


d^ 78970 


^ 13997 


62474 


59 




2 


66592 


-"51695 


^45038 


-^47391 


-^59634 


^82785 


^18024 


66738 


58 




3 


69613 


54847 


48335 


50845 


63261 


86603 


22055 


71007 


57 




4 


72637 


58003 


51634 


54301 


66891 


90424 


26089 


75279 


56 




5 


75662 


61160 


54935 


57760 


70524 


94249 


30127 


79556 


55 




6 


78690 


64320 


58239 


61222 


74160 


98077 


34169 


83837 


54 




7 


81720 


67482 


61545 


64687 


77798 


2601909 


38215 


88122 


53 




8 


84751 


70646 


64854 


68155 


81440 


05743 


42264 


92411 


52 




9 


87785 


73813 


68166 


71625 


85085 


09582 


46318 


96705 


51 




10 


90821 


76982 


71480 


75098 


88734 


13423 


50375 


12-3101003 


50 




11 


93860 


80153 


74797 


78574 


92385 


17269 


54435 


05305 


49 




12 


96900 


83327 


78116 


82052 


96039 


21117 


58500 


09611 


48 




13 


99942 


86503 


81437 


85534 


99696 


24969 


62568 


13922 


47 




14 


12-1602987 


89681 


84762 


89018 


2403357 


28824 


66641 


18237 


46 




15 


06034 


92861 


88088 


92505 


07020 


32683 


70717 


22556 


45 




16 


09082 


96044 


91418 


95995 


10687 


36545 


74797 


26880 


44 




17 


12133 


99230 


94749 


99487 


14356 


40411 


78880 


31208 


43 




18 


15187 


1802417 


98084 


2202983 


18029 


44280 


82968 


35540 


42 




19 


18242 


05607 


2001421 


06481 


21705 


48152 


87059 


39876 


41 




20 


21299 


08799 


04760 


09982 


25384 


52028 


91154 


44217 


40 




21 


24359 


11994 


08102 


13486 


29066 


55908 


95254 


48562 


39 




22 


27421 


15191 


11447 


16993 


32751 


59791 


99357 


52912 


38 




23 


30485 


18390 


14794 


20502 


36440 


63677 


2903463 


57266 


37 




24 


33551 


21592 


18144 


24015 


40131 


67567 


07574 


61624 


36 




25 


36619 


24796 


21497 


27530 


43826 


71460 


11689 


65987 


35 




26 


39689 


28002 


24852 


31048 


47523 


75357 


15807 


70354 


34 




27 


42762 


31211 


28209 


34569 


51224 


79258 


19930 


74725 


33 




28 


45837 


34422 


31569 


38093 


54928 


83161 


24056 


79101 


32 




29 


48913 


37636 


34932 


41619 


58636 


87069 


28187 


83481 


31 




30 


51993 


40852 


38298 


45149 


62346 


90980 


32321 


87866 


30 




31 


55074 


44070 


41666 


48681 


66060 


94894 


36459 


92255 


29 




32 


58157 


47291 


45036 


52216 


69776 


98812 


40601 


96649 


28 




33 


61243 


50514 


48410 


55754 


73496 


2702733 


44747 


12-3201047 


27 




34 


64331 


53739 


51785 


59295 


77220 


06658 


48897 


05449 


26 




35 


67421 


56967 


55164 


62839 


80946 


10587 


53051 


09856 


25 




36 


70513 


60197 


58545 


66386 


84675 


14519 


57209 


14267 


24 




37 


73608 


63430 


61929 


69936 


88408 


18455 


61371 


18683 


23 




38 


76704 


66665 


65315 


73488 


92144 


22394 


65537 


23104 


22 




39 


79803 


69902 


68704 


77044 


95883 


26337 


69707 


27529 


21 




40 


82904 


73142 


72096 


80602 


99626 


30283 


73881 


31958 


20 




41 


86008 


76385 


75490 


84164 


2503371 


34233 


78059 


36392 


19 




42 


89113 


79629 


78887 


87728 


07120 


38187 


82241 


40830 


18 




43 


92221 


82876 


82287 


91295 


10872 


42144 


86427 


45273 


17 




44 


95331 


86126 


85689 


94865 


14628 


46105 


90617 


49721 


16 




45 


98443 


89378 


89094 


98438 


18386 


50069 


94811 


54173 


15 




46 


12-1701557 


92632 


92502 


2302014 


22148 


54037 


99010 


58629 


14 




47 


04674 


95889 


95912 


05593 


25913 


58009 


3003212 


63091 


13 




48 


07793 


99149 


99325 


09175 


29681 


61984 


07418 


67557 


12 




49 


10914 


1902410 


2102741 


12760 


33453 


65963 


116-29 


72027 


11 




50 


14038 


05675 


06159 


16348 


37228 


69946 


15843 


76502 


10 




51 


17163 


08941 


09580 


19939 


41006 


73932 


20062 


80982 


9 




52 


20291 


12211 


13004 


23532 


44788 


77922 


24284 


85466 


8 




53 


23421 


15482 


16431 


27129 


48572 


81916 


28511 


89955 


7 




54 


26554 


18756 


19860 


30729 


52360 


85913 


32742 


94448 


6 




55 


29688 


22033 


23292 


34331 


56152 


89914 


36977 


98947 


5 




56 


32825 


25312 


26726 


37937 


59947 


93919 


41216 


12-3303449 


4 




57 


35964 


28593 


30164 


41546 


63745 


97927 


45459 


07957 


3 




58 


39106 


31877 


33604 


45157 


67546 


2801939 


49707 


12469 


2 




59 


42-250 


35164 


37046 


48772 


71351 


05955 


53958 


16986 


1 




60 


45396 


38453 


40492 


52390 


75159 


09974 


58214 


21508 







" 


23' i 


22' 


21' 


20' 


19' 


18' 


17' 


16' 


" 










I 


.OG. CO! 


'AN. 0° 






1 






















"■■■ 





{(1.56 LOG. SINE 89°. [Table ii. 




Il~ 


44' 


45' 


46' 


47' 


48' 


49' 


50' 


51' " i 




! 


9-9999953 


-9999959 


-9999964 


•9999969 


•9999974 


•9999978 


•9999982 


9^9999985 


60 




1 


53 


^ 59 


■^ 64 


^ 69 


as 74 


=^' 78 


=^ 82 


85 


59 




2 


53 


59 


64 


69 


74 


78 


82 


85 


58 1 




3 


53 


59 


64 


69 


74 


78 


82 


85 


57 




4 


53 


59 


64 


69 


74 


78 


82 


85 


56 




5 


53 


59 


64 


69 


74 


78 


82 


85 


55 




6 


54 


59 


64 


69 


74 


78 


82 


85 


54 




7 


54 


59 


65 


70 


74 


78 


82 


86 


53 




8 


54 


59 


65 


70 


74 


78 


82 


86 


52 




9 


9-9999954 


•9999959 


-9999965 


-9999970 


•9999974 


•9999978 


-9999982 


9-9999986 


51 




10 


54 


60 


65 


70 


74 


78 


82 


86 


50 




11 


54 


60 


65 


70 


74 


79 


82 


86 


49 




12 


54 


60 


65 


70 


74 


79 


82 


86 


48 




13 


54 


60 


65 


70 


74 


79 


82 


86 


47 




14 


54 


60 


65 


70 


75 


79 


■ 82 


86 


46 




15 


54 


60 


65 


70 


75 


79 


83 


86 


45 




16 


55 


60 


65 


70 


75 


79 


83 


86 


44 




17 


55 


60 


65 


70 


75 


79 


83 


86 


43 




18 


55 


60 


66 


70 


75 


79 


83 


86 


42 




19 


9-9999955 


-9999960 


-9999966 


-9999970 


•9999975 


-9999979 


•9999983 


9-9999986 


41 




20 


55 


60 


66 


71 


75 


79 


83 


86 


40 




21 


55 


61 


66 


71 


75 


79 


83 


86 


39 




22 


55 


61 


66 


71 


75 


79 


83 


86 


38 




23 


55 


61 


66 


71 


75 


79 


83 


86 


37 




24 


55 


61 


66 


71 


75 


79 


83 


86 


36 




25 


55 


61 


66 


71 


75 


79 


83 


86 


35 




26 


55 


61 


66 


71 


75 


79 


83 


87 


34 




27 


56 


61 


66 


71 


75 


80 


83 


87 


33 




28 


56 


61 


66 


71 


76 


80 


83 


87 


32 




29 


9-9999956 


•9999961 


-9999966 


-9999971 


•9999976 


•9999980 


•QOQQQOQ 

nyyyyoo 


9-9999987 


31 




30 


56 


61 


67 


71 


76 


80 


83 


87 


30 




31 


56 


61 


67 


71 


76 


80 


83 


87 


29 




32 


56 


62 


67 


71 


76 


80 


84 


87 


28 




33 


56 


62 


67 


72 


76 


80 


84 


87 


27 




34 


56 


62 


67 


72 


76 


80 


84 


87 


26 




35 


56 


62 


67 


72 


76 


80 


84 


87 


25 




36 


56 


62 


67 


72 


76 


80 


84 


87 


24 




37 


57 


92 


67 


72 


76 


80 


84 


87 


23 




38 


57 


62 


67 


72 


76 


80 


84 


87 


22 




39 


9-9999957 


-9999962 


•9999967 


•9999972 


•9999976 


-9999980 


•9999984 


9-9999987 


21 




40 


57 


62 


67 


72 


76 


80 


84 


87 


20 




41 


57 


62 


67 


72 


76 


80 


84 


87 


19 




42 


57 


62 


67 


72 


77 


81 


84 


87 


18 




43 


57 


63 


68 


72 


77 


81 


84 


87 


17 




44 


57 


63 


68 


72 


77 


81 


84 


87 


16 




45 


57 


63 


68 


72 


77 


81 


84 


87 


15 




46 


57 


63 


68 


73 


77 


81 


84 


88 


14 




47 


57 


63 


68 


73 


77 


81 


84 


88 


13 




48 


58 


63 


68 


73 


77 


81 


84 


88 


12 




49 


9-9999958 


•9999963 


-9999968 


•9999973 


-9999977 


•9999981 


•9999985 


9-9999988 


11 




50 


58 


63 


68 


73 


77 


81 


85 


88 


10 




51 


58 


63 


68 


73 


77 


81 


85 


88 


9 




52 


58 


63 


68 


73 


77 


81 


85 


88 


8 




53 


58 


63 


68 


73 


77 


61 


85 


88 


7 




54 


58 


63 


68 


73 


77 


81 


85 


88 


6 




55 


58 


64 


69 


73 


77 


81 


85 


88 


5 




56 


58 


64 


69 


73 


77 


81 


85 


88 


4 




57 


58 


64 


69 


73 


78 


81 


85 


88 


3 




58 


58 


64 


69 


73 


78 


82 


85 


88 


2 




i59 


9-9999959 


•9999964 


-9999969 


^9999973 


-9999978 


•9999982 


9999985 


9-9999988 


1 




160 


59 


64 


69 


74 


78 


82 


85 


88 







// 


15' 


14' 


13' 


12' 


11' 


10' 


9' 


8' 


// 




LOG. COSINE 0°. ( 





[Table ii.] log. tan. 89°. 


157 




" 


44' 


45' 


46' 


47' 


48' 


49' 


50' 


51' 


// 







12-3321508 


3601799 


3901434 


4223285 


4570909 


4948797 


5362727 


12-5820304 


60 




1 


26034 


A, 06627 


^06608 


<^ 28856 


d^ 76945 


^ 55382 


A. 69970 


28354 


59 




2 


30565 


^11461 


^11787 


^ 34435 


^82989 


^61978 


"^ 77227 


36419 


58 




3 


35101 


16300 


16973 


40021 


89042 


68583 


84496 


44499 


57 




4 


39641 


21144 


22165 


45614 


95104 


75198 


91777 


52594 


56 




5 


44187 


25994 


27363 


51214 


4601174 


81824 


99070 


60704 


55 




6 


48787 


30849 


32567 


56821 


07252 


88459 


5406375 


68829 


54 




7 


53291 


35710 


37778 


62436 


13339 


95105 


13692 


76970 


53 




8 


57851 


40576 


42995 


68058 


19434 


5001761 


21022 


85125 


52 




9 


62415 


45447 


48218 


73687 


25538 


08427 


28365 


93297 


51 




10 


66985 


50324 


53447 


79324 


31651 


15103 


35719 


12-5901483 


50 




11 


71559 


55207 


58683 


84968 


37772 


21790 


43087 


09685 


49 




12 


76137 


60095 


63925 


90619 


43902 


28487 


50466 


17903 


48 




13 


80721 


64988 


69173 


96278 


50040 


35194 


57859 


26136 


47 




14 


85310 


69887 


74428 


4301944 


56187 


41912 


65263 


34384 


46 




15 


89903 


74792 


79689 


07617 


62343 


48640 


72681 


42649 


45 




16 


94501 


79702 


84956 


13298 


68508 


55379 


80111 


50929 


44 




17 


99104 


84618 


90230 


18986 


74681 


62128 


87554 


59225 


43 




18 


12-3403712 


89539 


95510 


24682 


80863 


68887 


95010 


67537 


42 




19 


08325 


94466 


4000797 


30385 


87054 


75657 


5502479 


75865 


41 




20 


12943 


99398 


06090 


36096 


93254 


82438 


09960 


84209 


40 




21 


17565 


3704336 


11389 


41814 


99463 


89229 


17454 


92569 


39 




22 


22193 


09280 


16696 


47540 


4705681 


96031 


24962 


12-6000945 


38 




23 


26826 


14229 


22008 


53273 


11907 


5102843 


32482 


09337 


37 




24 


31463 


19184 


27327 


59014 


18142 


09666 


40015 


17745 


36 




25 


36105 


24145 


32653 


64762 


24387 


16500 


47562 


26170 


35 




26 


40753 


29111 


37985 


70519 


30640 


23345 


.55121 


34611 


34 




27 


45405 


34083 


43323 


76282 


36903 


30201 


62694 


43069 


33 




28 


50053 


39061 


48669 


82054 


43174 


37067 


70280 


51543 


32 




29 


54725 


44044 


54020 


87833 


49455 


43944 


77879 


60033 


31 




30 


59392 


49033 


59379 


93620 


55744 


50832 


85492 


68541 


30 




31 


64065 


54028 


64744 


99414 


62043 


57731 


93118 


77065 


29 




32 


68742 


59028 


70116 


1405216 


68351 


64641 


5600757 


85605 


28 




33 


73425 


64035 


75494 


11026 


74668 


71563 


08410 


94163 


27 




34 


78112 


69047 


80879 


16844 


80994 


78495 


16076 


12-6102737 


26 




35 


82805 


74065 


86270 


22670 


87330 


85438 


23756 


11329 


25 




36 


87503 


79089 


91669 


28503 


93674 


92392 


31449 


19937 


24 




37 


92205 


84118 


97074 


34344 


4800028 


99358 


39157 


28563 


23 




38 


96913 


89153^ 


1102486 


40194 


06392 


5206334 


46877 


37206 


22 




39 


12-3501626 


94195 


07904 


46051 


12764 


13322 


54612 


45866 


21 




40 


06344 


99242 


13330 


51916 


19146 


20321 


62360 


54543 


20 




41 


11067 ' 


J804295 


18762 


57788 


25538 


27332 


70123 


63237 


19 




42 


15796 


09353 


24201 


63669 


31939 


34354 


77899 


71949 


18 




43 


20529 


14418 


29647 


69558 


38349 


41387 


85689 


80679 


17 




44 


25268 


19489 


35099 


75455 


44769 


48431 


93493 


89426 


16 




45 


30012 


24565 


40559 


81360 


51198 


55487 


5701311 


98191 


15 




46 


34761 


29648 


46025 


87272 


57637 


62555 


09143 


12-6206974 


14 




47 


39515 


34736 


51498 


93193 


64085 


69634 


16990 


15774 


13 




48 


44275 


39831 


56978 


99122 


70543 


76724 


24850 


24592 


12 




49 


49039 


44931 


62465 


1505059 


77011 


B382V 


32725 


33428 


11 




50 


53809 


50037 


67959 


11005 


83488 


90940 


40614 


42282 


10 




51 


58584 


55150 


73460 


16958 


89975 


98066 


48518 


51155 


9 




52 


63365 


60268 


78968 


22920 


96472 


5305203 


56436 


60045 


8 




53 


68150 


65393 


84483 


28889 


4902978 


12352 


64368 


68954 


7 




54 


72941 


70523 


90005 


34867 


09494 


19513 


72315 


77881 


6 




55 


77738 


75660 


95534 


40853 


16020 


26685 


80276 


86826 


5 




56 


82539 


80803 ^ 


1201070 


46848 


22556 


33870 


88252 


95790 


4 




57 


87346 


85951 


06613 


52851 


29101 


41066 


96243 


12-6304772 


3 




58 


92158 


91106 


12163 


58862 


35657 


48274 


5804248 


13773 


2 




59 


96976 


96267 


17720 


64881 


42222 


55494 


12269 


22793 


1 




60 


12-3601799 


J901434 


23285 


70909 


48797 


62727 


20304 


31831 







// 


15' 


14' 


13' 


12' 


11' 


10' 


9' 


8' 


" 




LmB 


LOG. COTAN. 0°. 


\ 





14 





158 LOG. SINE 89°. [Table 11 


n 




'■ 


52' 


53' 


W 1 55' 1 


56' 


57' 


58' 


59' 1 


// 







9-9999988 


9999991 


9999993 


9999995 


9999997 


9999998 


9-9999999 


10-OCOOOOO 


oO 




1 


88 


^ 91 


^ 93 


^ 95 


^ 97 < 


=^ 98 


99 


CO 


59 




2 


88 


91 


93 


95 


97 


98 


d9 


00 


58 




3 


88 


91 


93 


95 


97 


98 


99 


00 


57 




4 


88 


91 


94 


95 


97 


98 


99 


00 


56 




5 


89 


91 


94 


96 


97 


98 


99 


00 


55 




6 


89 


91 


94 


96 


97 


98 


99 


00 


54 




7 


89 


91 


94 


96 


97 


98 


99 


00 


53 




8 


89 


91 


94 


96 


97 


98 


99 


00 


52 




9 


9-9999989 


-9999991 


9999994 


9999996 


9999997 


9999999 


9-9999999 


10-0000000 


51 




10 


89 


91 


94 


96 


97 


99 


99 


00 


50 




11 


89 


91 


94 


96 


97 


99 


99 


00 


49 




12 


89 


92 


94 


96 


97 


99 


99 


00 


48 




13 


89 


92 


94 


96 


97 


99 


99 


00 


47 




14 


89 


92 


94 


96 


97 


99 


99 


00 


46 




15 


89 


92 


94 


96 


97 


99 


99 


00 


45 




16 


89 


92 


94 


96 


97 


99 


99 


00 


44 




17 


89 


92 


94 


96 


97 


99 


99 


00 


43 




18 


89 


92 


94 


96 


97 


99 


99 


00 


42 




19 


9-9999989 


•9999992 


•9999994 


•9999996 


-9999993 


-9999999 


9-9999999 


10-0000000 


41 




20 


89 


92 


94 


96 


98 


99 


99 


00 


40 




21 


89 


92 


94 


96 


98 


99 


99 


00 


39 




22 


89 


92 


94 


96 


98 


99 


10-0000000 


0013811 




23 


89 


92 


94 


96 


98 


99 


00 


00!37|| 




24 


89 


92 


94 


96 


98 


99 


00 


00 


36 




25 


90 


92 


94 


96 


98 


99 


00 


00 


35 




26 


90 


92 


94 


96 


98 


99 


00 


00 


34 




27 


90 


92 


94 


96 


98 


99 


00 


00 


33 




28 


90 


92 


94 


96 


98 


99 


00 


00 


32 




29 


9-9999990 


•9999992 


•9999994 


-9999996 


-9999998 


•9999999 


10-0000000 


10^0000000 


31 




30 


90 


92 


94 


96 


98 


99 


00 


00 


30 




31 


90 


92 


94 


96 


98 


99 


00 


00 


29 




32 


90 


92 


95 


96 


98 


99 


00 


00 


28 




33 


90 


92 


95 


96 


98 


99 


00 


00 


27 




34 


90 


92 


95 


96 


98 


99 


00 


00 


26 




35 


90 


92 


95 


96 


98 


99 


00 


00 


25 




36 


90 


92 


95 


96 


98 


99 


00 


00 


24 




37 


90 


93 


95 


96 


98 


99 


00 


00 


23 




38 


90 


93 


95 


96 


98 


99 


00 


00 


22 




39 


9-9999990 


•9999993 


•9999995 


•9999997 


•9999998 


-9999999 


10-0000000 


10^0000000 


21 




40 


90 


93 


95 


97 


98 


99 


00 


00 


20 




41 


90 


93 


95 


97 


98 


99 


00 


00 


19 




42 


90 


93 


95 


97 


98 


99 


00 


00 


18 




43 


90 


93 


95 


97 


98 


99 


00 


00 


17 




44 


90 


93 


95 


97 


98 


99 


00 


00 


16 




45 


90 


93 


95 


97 


98 


99 


00 


00 


15 




46 


90 


93 


95 


97 


98 


99 


00 


00 


14 




47 


90 


93 


95 


97 


98 


99 


00 


00 


13 




48 


90 


93 


95 


97 


98 


99 


00 


00 


12 




49 


9-9999991 


•9999993 


•9999995 


•9999997 


-9999998 


-9999999 


10-0000000 


10^0000000 


U 




50 


91 


93 


95 


97 


98 


99 


00 


00 


10 




51 


91 


93 


95 


97 


98 


99 


00 


00 


9 




52 


91 


93 


95 


97 


98 


99 


00 


00 


8 




53 


91 


93 


95 


97 


98 


99 


00 


00 


7 




54 


91 


83 


95 


97 


98 


99 


00 


00 


6 




55 


91 


93 


95 


97 


98 


99 


GO 


00 


5 




56 


91 


93 


95 


97 


98 


99 


00 


00 


4 




57 


91 


93 


95 


97 


98 


99 


00 


00 


3 




58 


91 


93 


95 


97 


98 


99 


00 


00 


2 




59 


9-9999991 


•9999993 


-9999995 


•9999997 


•9999998 


-9999999 


10-0000000 


10 0000000 


1 




60 


91 


93 


95 


97 


98 


99 


00 


00 







" 


1 r 


6' 


5' 


4' 


3' 


2' 


V 


0' 


'''' 




LOG. COSINE 0°. Ij 



Table ii.J log. 


TAN. 89°. 






159 II 


// 


52' 


53' 


54' 


55' 


56' 


57' 


58' 


59' 


/' 





12-6331831 


6911752 


7581222 


8373036 


13-9342137 


13-0591525 13"2352438 


135362739 


60 


1 


40888 


^22105 


93302 


87536 


360270 


615720 


388781 


5435731 


59 


2 


49965 


^32483 


7605417 


8402086 


378480 


640050 


425430 


5509971 


58 


3 


59060 


42885 


^ 17565 


^ 16684 


396766 


664518 


462392 


5585503 


57 


4 


68174 


53312 


""29747 


"^ 31331 


415129 


689124 


499671 


5662371 


56 


5 


77308 


63765 


41964 


46028 


433571 


713870 


537272 


5740624 


55 


6 


86460 


74242 


54214 


60775 


452091 


738758 


575202 


5820314 


54 


7 


95632 


84745 


66500 


75572 


470690 


763789 


613466 


5901493 


53 


8 


12-6404824 


95273 


78820 


90420 


489370 


788966 


652071 


5984218 


52 


9 


14035 


7005827 


91176 


8505319 


508130 


814289 


691021 


6068550 


51 


10 


23265 


16407 


7703567 


20268 


526971 


839761 


730324 


6154551 


50 


11 


32515 


27013 


15993 


35270 


545895 


865383 


769986 


6242291 


49 


12 


41785 


37644 


28455 


50324 


564901 


891158 


810013 


6331839 


48 


13 


51075 


48302 


40952 


65430 


583991 


917086 


850413 


6423273 


47 


14 


60385 


58985 


53486 


80588 


603165 


943170 


891192 


6516673 


46 


15 


69714 


69096 


66056 


95800 


622424 


969411 


932358 


6612126 


45 


16 


79064 


80432 


78663 


8611065 


641769 


995812 


973918 


6709725 


44 


17 


88434 


91195 


91306 


26384 


661201 


13-1022374 


13-3015879 


6809567 


43 


IS 


97824 


7101985 


7803986 


41758 


680720 


049100 


058249 


6911758 


42 


19 


12-6507235 


12802 


16704 


57185 


700327 


075992 


101037 


7016413 


41 


20 


16666 


23646 


29458 


72668 


720023 


103051 


144251 


7123651 


40 


21 


26117 


34517 


42250 


88207 


739809 


130279 


187899 


7233605 


39 


22 


35589 


45415 


55080 


8703801 


759685 


157680 


231990 


7346415 


38 


23 


45082 


56341 


67948 


19451 


779652 


185254 


276534 


7462234 


37 


24 


54596 


67294 


80855 


35158 


799712 


213005 


321539 


7581226 


36 


25 


64130 


78275 


93799 


50922 


819865 


240934 


367015 


7703571 


35 


26 


73686 


89284 


7906783 


66743 


840112 


269043 


412973 


7829462 


34 


27 


83262 


7200321 


19805 


82622 


860454 


297336 


459422 


7959112 


33 


28 


92860 


11385 


32867 


98560 


880891 


325815 


506373 


8092752 


32 


29 


11-6602479 


22479 


45968 


8814556 


901425 


354481 


553837 


8230634 


31 


30 


12119 


33600 


59108 


30611 


922057 


383338 


601826 


8373039 


30 


31 


21781 


44750 


72289 


46726 


942787 


412388 


650351 


8520271 


29 


32 


31464 


55929 


85509 


62901 


963617 


441633 


699424 


8672671 


28 


33 


41169 


67137 


98770 


79136 


984546 


471077 


749059 


8830614 


27 


34 


50896 


78373 


8012072 


95433 


13-0005578 


500722 


799267 


8994518 


26 


35 


60644 


89639 


25414 


8911790 


026711 


530571 


850062 


9164851 


25 


36 


70415 


7300934 


38798 


28210 


047948 


560626 


901458 


9342139 


24 


37 


80207 


12259 


52223 


44691 


069290 


590890 


953470 


9526973 


23 


38 


90022 


23613 


65689 


61236 


090736 


621367 


13-4006113 


9720025 


22 


39 


99859 


34997 


79198 


77844 


112289 


652060 


059401 


9922058 


21 


40 


12-6709718 


46410 


92748 


94516 


133950 


682970 


113351 


14-0133951 


20 


41 


09600 


57854 


8106341 


9011251 


155719 


714103 


167980 


0356715 


19 


42 


29504 


69328 


19977 


28052 


177598 


745460 


223305 


0591526 


18 


43 


39430 


80833 


33665 


44918 


199588 


777045 


279344 


0839762 


17 


44 


49380 


92368 


47377 


61849 


221689 


808862 


336115 


1103052 


16 


45 


59352 


7403934 


61142 


78847 


243904 


840913 


393639 


1383339 


15 


46 


69348 


15531 


74951 


95912 


266233 


873203 


451934 


1682971 


14 


47 


79366 


27158 


88805 


9113044 


288677 


905734 


511023 


2004818 


13 


48 


89408 


38817 


8202702 


30244 


311238 


938511 


570926 


2352439 


12 


49 


99472 


50508 


16644 


47512 


333916 


971538 


631668 


2730324 


11 


50 


12-6809560 


62229 


30631 


64849 


356714 


13-2004817 


693271 


3144251 


10 


51 


19672 


73983 


44663 


82256 


379632 


038354 


755760 


3601826 


9 


52 


29807 


85769 


58741 


99732 


402672 


072151 


819162 


4113351 


8 


53 


39966 


97586 


72864 


9217280 


425834 


106214 


883503 


4693271 


7 


54 


50149 


7509436 


87034 


34898 


449121 


140545 


948812 


5362739 


6 


55 


60356 


21318 


8301250 


52588 


472533 


175151 


13-5015118 


6154551 


5 


56 


70587 


33233 


15512 


70351 


496072 


210034 


082451 


7123651 


4 


57 


80842 


45181 


29822 


88187 


519739 


245200 


150846 


8373039 


3 


58 


91121 


57161 


44179 


9306096 


543536 


280653 


220334 


15-0133951 


2 


59 


12-6901424 


69175 


58583 


24079 


567464 


316397 


290953 


3144251 


1 


60 


11752 


81222 


73036 


42137 


591525 


352438 


362739 


H-cD 





" 


7' 


6' 


5' 


4' 


3' 


2' 


1' 


0' " 


_ 


LOG. C 


DTAN. 0°. 






^1 



NATURAL SIGNS AND TANGENTS 



TO EVERY DEGREE AND MINUTE OF THE QUADRANT. 



|162 NAT. SINE. [Table in7l 


' 


0° 


1 1° 


1 2° 


3° 


1 4° 


5° 


6° 


70 


/ 





000 0000 


017 4524 


'034 8995 


052 3360 


'069 7565 


087 1557 


104 5285 


121 8693 


60 


1 


2909 


7432 


;035 1902 


6264 


070 0467 


4455 


8178 


122 1581 


59 


2 


581S 


018 0341 


4809 


9169 


3368 


7353 


105 1070 


4468 


58 


3 


8727 


3249 


7716 


053 2074 


6270 


088 0251 


3963 


7355 


57 


4 


001 1636 


6158 


036 0623 


4979 


9171 


3148 


6856 


123 0241 


56 


5 


4544 


9066 


3530 


7883 


071 2073 


6046 


9748 


3128 


55 


6 


7453 


019 1974 


6437 


054 0788 


4974 


8943 


106 2641 


6015 


54 


7 


002 0362 


4833 


9344 


3693 


7876 


089 1840 


5533 


8901 


53 


8 


3271 


7791 


037 2251 


6597 


072 0777 


4738 


8425 


124 1788 


52 


9 


6180 


020 0699 


5158 


9502 


3678 


7635 


107 1318 


4674 


51 


10 


9089 


3608 


8065 


055 2406 


6530 


090 0532 


4210 


7560 


50 


11 


003 1998 


6516 


038 0971 


5311 


9481 


3429 


7102 


125 0446 


49 


12 


4907 


9424 


3878 


8215 


073 2382 


6326 


9994 


3332 


48 


13 


7815 


021 2332 


6785 


056 1119 


5283 


9223 


108 2885 


6218 


47 


14 


004 0724 


5241 


9692 


4024 


8184 


091 2119 


5777 


9104 


46 


15 


3633 


8149 


039 2598 


6928 


074 1085 


5016 


8669 


126 1990 


45 


16 


6542 


022 1057 


5505 


9832 


3986 


7913 


109 1560 


4875 


44 


17 


9451 


3965 


8411 


057 2736 


6887 


092 0809 


4452 


7761 


43 


18 


005 2360 


6873 


040 0318 


5640 


9787 


3706 


7343 


127 0646 


42 


19 


5268 


9781 


4224 


8544 


075 2688 


6602 


110 0234 


3531 


41 


20 


8177 


023 2690 


7131 


058 1448 


5589 


9499 


3126 


6416 


40 


21 


006 1086 


5598 


041 0037 


4352 


8489 


093 2395 


6017 


9302 


39 


22 


3995 


8506 


2944 


7256 


076 1390 


5291 


8908 


128 2186 


38 


23 


6904 


024 1414 


5850 


059 0160 


4290 


8187 


111 1799 


5071 


37 


24 


9813 


4322 


8757 


3064 


7190 


094 1083 


4689 


7956 


36- 


25 


007 2721 


7230 


042 1663 


5967 


077 0091 


3979 


7580 


129 0841 


35 


26 


5630 


025 0138 


4569 


8871 


2991 


6875 


112 0471 


3725 


34 


27 


8539 


3046 


7475 


060 1775 


5891 


9771 


3361 


6609 


33 


28 


008 1448 


5954 


043 0382 


4678 


8791 


095 2666 


6252 


9494 


32 


29 


4357 


8862 


3288 


7582 


078 1691 


5562 


9142 


130 2378 


31 


30 


7265 


026 1769 


6194 


061 0485 


4591 


8458 


113 2032 


5262 


30 


31 


009 0174 


4677 


9100 


3389 


7491 


096 1353 


4922 


8146 


29 


32 


3083 


7585 


044 2006 


6292 


079 0391 


4248 


7812 


131 1030 


28 


33 


5992 


027 0493 


4912 


9196 


3290 


7144 


114 0702 


3913 


27 


34 


8900 


3401 


7818 


062 2099 


6190 


097 0039 


3592 


6797 


26 


35 


010 1809 


6309 


045 0724 


5002 


9090 


2934 


6482 


9681 


25 


36 


4718 


9216 


3630 


7905 


080 1989 


5829 


9372 


132 2564 


24 


37 


7627 


028 2124 


6536 


063 0808 


4389 


8724 


115 2261 


5447 


23 


38 


Oil 0535 


5032 


9442 


3711 


7788 


098 1619 


5151 


8330 


22' 


39 


3444 


7940 


046 2347 


6614 


081 0687 


4514 


8040 


133 1213 


21 


40 


6353 


029 0847 


5253 


9517 


3587 


7408 


116 0929 


4096 


20 


41 


9261 


3755 


8159 


064 2420 


6486 


099 0303 


3818 


6979 


19 


42 


012 2170 


6662 


047 1065 


5323 


9385 


3197 


6707 


9862 


18 


43 


5079 


9570 


3970 


8226 


082 2284 


6092 


9596 


134 2744 


17 


44 


7987 


030 2478 


6876 


065 1129 


5183 


8986 


117 2485 


5627 


16 


45 


013 0896 


5385 


9781 


4031 


8082 


100 1881 


5374 


8509 


15 


46 


3805 


8293 


048 2687 


6934 


083 0981 


4775 


8263 


135 1392 


14 


47 


6713 


031 1200 


5592 


9836 


3880 


7669 


118 1151 


4274 


13 


48 


9622 


4108 


8498 


066 2739 


6778 


101 0563 


4040 


7156 


12 


49 


014 2530 


7015 


049 1403 


5641 


9677 


3457 


6928 


136 0038 


11 


50 


5439 


9922 


4308 


8544 


084 2576 


6351 


9816 


2919 


10 


51 


8348 


032 2830 


7214 


067 1446 


5474 


9245 


119 2704 


5801 


9 


52 


015 1256 


5737 


050 0119 


4349 


8373 


102 2138 


5593 


8683 


8 


53 


4165 


8644 


3024 


7251 


085 1271 


5032 


8431 


137 1564 


7 


54 


7073 


033 1552 


5929 


068 0153 


4169 


7925 


120 1368 


4445 


6 


55 


9982 


4459 


8835 


3055 


7067 


103 0319 


4256 


7327 


5 


56 


016 2890 


7366 


051 1740 


5957 


9966 


3712 


7144 


138 0208 


4 


57 


5799 


034 0274 


4645 


8859 


086 2364 


6605 


121 0031 


3089 


3 


58 


8707 


3181 


7550 


069 1761 


5762 


9499 


2919 


5970 


2 


59 


017 1616 


6088 


052 0455 


4663 


8660 


104 2392 


5806 


8850 


1 1 


60 


4524 


8995 


3360 


7565 


087 1557 


5235 


8693 


139 1731 


1 


' 


89° 


88° 


87° 


86° 


85° 


84° 


83° 


820 


' 1 


NAT. COSINE. || 



Table iii.] nat. tan. 163 




/ 


0° 


1° 


2o 


30 


40 


50 


6° j 7° 


/ 







000 0000 


017 4551 


034 9208 


052 4078 


069 9268 


087 4887 


105 1042 


122 7846 


60 




1 


2909 


7460 


035 2120 


6995 


070 2191 


7818 


3983 


123 0798 


59 




2 


581S 


018 0370 


5033 


9912 


5115 


088 0749 


6925 


3752 


58 




3 


8727 


3280 


7945 


053 2829 


8038 


3681 


9866 


6705 


57 




4 


001 1636 


6190 


036 0858 


5746 


071 0961 


6612 


106 2808 


9658 


56 




5 


4544 


9100 


3771 


8663 


3885 


9544 


5750 


124 2612 


55 




6 


7453 


019 2010 


6683 


054 1581 


6809 


089 2476 


8692 


5566 


54 




7 


002 0362 


4920 


9596 


4498 


9733 


5408 


107 1634 


8520 


53 




8 


3271 


7830 


037 2509 


7416 


072 2657 


8341 


4576 


125 1474 


52 




9 


618U 


020 0740 


5422 


055 0333 


5581 


090 1273 


7519 


4429 


51 




10 


9089 


3650 


8335 


3251 


8505 


4206 


108 0462 


7384 


50 




11 


003 1998 


6560 


038 1248 


6169 


073 1430 


7138 


3405 


126 0339 


49 




12 


4907 


9470 


4161 


9087 


4354 


091 0071 


6348 


3294 


48 




13 


7816 


021 2380 


7074 


056 2005 


7279 


3004 


9291 


6249 


47 




14 


004 0725 


5291 


9988 


4923 


074 0203 


5938 


109 2234 


9205 


46 




15 


3634 


8201 


039 2901 


7841 


3128 


8871 


5178 


127 2161 


45 




16 


6542 


022 1111 


5814 


057 0759 


6053 


092 1804 


8122 


5117 


44 




17 


9451 


4021 


8728 


3678 


8979 


4738 


110 1066 


8073 


43 




18 


005 2360 


6932 


040 1641 


6596 


075 1904 


7672 


4010 


128 1030 


42 




19 


5269 


9842 


4555 


9515 


4829 


093 0606 


6955 


3986 


41 




20 


8178 


023 2753 


7469 


058 2434 


7755 


3540 


9899 


6943 


40 




21 


006 1087 


6663 


041 0383 


5352 


076 0680 


6474 


111 2844 


9900 


39 




22 


3996 


8574 


3296 


8271 


3606 


9409 


5789 


129 2858 


38 




23 


6905 


024 1484 


6210 


059 1190 


6532 


094 2344 


8734 


5815 


37 




24 


9814 


4395 


9124 


4109 


9458 


5278 


112 1680 


8773 


36 




25 


007 2723 


7305 


042 2038 


7029 


077 2384 


8213 


4625 


130 1731 


35 




26 


5632 


025 0216 


4952 


9948 


5311 


095 1148 


7571 


4690 


34 




27 


8541 


3127 


7866 


060 2867 


8237 


4084 


113 0517 


7648 


33 




28 


008 1450 


6038 


043 0781 


5787 


078 1164 


7019 


3463 


131 0607 


32 




29 


4360 


8948 


3695 


8706 


4090 


9955 


6410 


3566 


31 




30 


7269 


026 1859 


6609 


061 1626 


7017 


096 2890 


9356 


6525 


30 




31 


009 0178 


4770 


9524 


4546 


9944 


5826 


114 2303 


9484 


29 




32 


3087 


7681 


044 2438 


7466 


079 2871 


8763 


5250 


132 2444 


28 




33 


5996 


027 0592 


5353 


062 0386 


5798 


097 1699 


8197 


5404 


27 




34 


8905 


3503 


8268 


3306 


8726 


4635 


115 1144 


8364 


26 




35 


010 1814 


6414 


045 1183 


6226 


080 1653 


7572 


4092 


133 1324 


25 




36 


4724 


9325 


4097 


9147 


4581 


098 0509 


7039 


4285 


24 




37 


7633 


028 2236 


7012 


063 2067 


7509 


3446 


9987 


7246 


23 




38 


Oil 0542 


5148 


9927 


4988 


081 0437 


6383 


116 2936 


134 0207 


22 




39 


3451 


8059 


046 2842 


7908 


3365 


9320 


5884 


3168 


21 




40 


6361 


029 0970 


5757 


064 0829 


6293 


099 2257 


8832 


6129 


20 




41 


9270 


3882 


8673 


3750 


9221 


5194 


117 1781 


9091 


19 




42 


012 2179 


6793 


047 1588 


6671 


082 2150 


8133 


4730 


135 2053 


18 




43 


5088 


9705 


4503 


9592 


5078 


100 1071 


7679 


5015 


17 




44 


7998 


030 2616 


7419 


065 2513 


8007 


4009 


118 0628 


7978 


16 




45 


013 0907 


5528 


048 0334 


5435 


083 0936 


6947 


3578 


136 0940 


15 




46 


3817 


8439 


3250 


8356 


3865 


9886 


6528 


3903 


14 




47 


6726 


031 1351 


6166 


066 1278 


6794 


101 2824 


9478 


6866 


13 




48 


9635 


4263 


9082 


4199 


9723 


5763 


119 2428 


9830 


12 




49 


014 2545 


7174 


049 1997 


7121 


084 2653 


8702 


5378 


137 2793 


11 




50 


5454 


032 0086 


4913 


067 0043 


5583 


102 1641 


8329 


5757 


10 




51 


8364 


2998 


7829 


2965 


8512 


4580 


120 1279 


8721 


9 




52 


015 1273 


5910 


050 0746 


5887 


085 1442 


7520 


4230 


138 1685 


8 




53 


4183 


8822 


3662 


8809 


4372 


103 0460 


7182 


4650 


7 




54 


7093 


033 1734 


6578 


068 1732 


7302 


3399 


121 0133 


7615 


6 




55 


016 0002 


4646 


9495 


4654 


086 0233 


6340 


3085 


139 0580 


5 




56 


2912 


7558 


051 2411 


7577 


3163 


9280 


6036 


3545 


4 




57 


5821 


034 0471 


5328 


069 0499 


6094 


104 2220 


8988 


6510 


3 




58 


8731 


3383 


8244 


3422 


9025 


5161 


122 1941 


9476 


2 




59 


017 1641 


6295 


052 1161 


6345 


087 1956 


8101 


4893 


140 2442 


1 




60 


4551 


9208 


4078 


9268 


4887 


105 1042 


7846 


5408 







/ 


89° 


880 


87° 


86° 


85° 


840 


83° 


820 






_ NAT. COTAN. | 





h64 NAT. SINE. [ TaWe in.jl 




/ 


8° 


9° 


10° 


11° 


12° 


13° 


14° 1 


15° 


/ 







139 1731 


156 4345 


173 6482 


190 8090 


207 9117 


224 9511 241 9219| 


258 8190 


60 




1 


4612 


7218 


9346 


191 0945 


208 1962 


225 2345 


242 2041 


259 1000 


59 




2 


7492 


157 0091 


174 2211 


3801 


4807 


5179 


4863 


3810 


58 I 




3 


140 0372 


2963 


5075 


6656 


7652 


8013 


7685 


6619 


57' 




4 


3252 


5836 


7939 


9510 


209 0497 


226 0846 


243 0507 


9428 


56 




5 


6132 


8708 


175 0803 


192 2365 


3341 


3680 


3329 


260 2237 


55 




6 


9012 


158 1581 


3667 


5220 


6186 


6513 


6150 


5045 


54 




7 


141 1892 


4453 


6531 


8074 


9030 


9346 


8971 


7853 


53 




8 


4772 


7325 


9395 


193 0928 


210 1874 


227 2179 


244 1792 


261 0662 


52 




9 


7651 


159 0197 


176 2258 


3782 


4718 


5012 


4613 


3469 


51 




10 


142 0531 


3069 


5121 


6636 


7561 


7844 


7433 


6277 


50 




11 


3410 


5940 


7984 


9490 


211 0405 


228 0677 


245 0254 


9085 


49 




12 


6289 


8812 


177 0847 


194 2344 


3248 


3509 


3074 


262 1892 


48 




13 


9168 


160 1683 


3710 


5197 


6091 


6341 


5894 


4699 


47 




14 


143 2047 


4555 


6573 


8050 


8934 


9172 


8713 


7506 


46 1 




15 


4926 


7426 


9435 


195 0903 


212 1777 


229 2004 


246 1533 


263 0312 


45' 




16 


7805 


161 0297 


178 2298 


3756 


4619 


4835 


4352 


3118 


44 




17 


144 0684 


3167 


5160 


6609 


7462 


7666 


7171 


5925 


43: 




18 


3562 


6038 


8022 


9461 


213 0304 


230 0497 


9990 


8730 


42! 




19 


6440 


8909 


179 0884 


196'2314 


3146 


3328 


247 2809 


264 1536 


41 




20 


9319 


162 1779 


3746 


5166 


5988 


6159 


5627 


4342 


40 




21 


145 2197 


4650 


6607 


8018 


8829 


8989 


8445 


7147 


39 




22 


5075 


7520 


9469 


197 0870 


214 1671 


231 1819 


248 1263 


9952 


38 




23 


7953 


163 0390 


180 2330 


3722 


4512 


4649 


4081 


265 2757 


37 




24 


146 0830 


3260 


5191 


6573 


7353 


7479 


6899 


5561 


36 




25 


3708 


6129 


8052 


9425 


215 0194 


232 0309 


9716 


8366 


35 




26 


6585 


8999 


181 0913 


198 2276 


3035 


3138 


249 2533 


266 1170 


34 




27 


9463 


164 1868 


3774 


5127 


5876 


5967 


5350 


3973 


33 




28 


147 2340 


4738 


6635 


7978 


8716 


8796 


8167 


6777 


32 




29 


5217 


7607 


9495 


199 0829 


216 1556 


233 1625 


250 0984 


9581 


31 




30 


8094 


165 0476 


182 2355 


3679 


4396 


4454 


3800 


267 2384 


30 




31 


148 0971 


3345 


5215 


6530 


7236 


7282 


6616 


5187 


29 




32 


3848 


6214 


8075 


9380 


217 0076 


234 0110 


9432 


7989 


28 




33 


6724 


9082 


183 0935 


200 2230 


2915 


2938 


251 2248 


268 0792 


27 




34 


9601 


166 1951 


3795 


5080 


5754 


5766 


5063 


3594 


26 




35 


149 2477 


4819 


6654 


7930 


8593 


8594 


7879 


6396 


25 




36 


5353 


7687 


9514 


201 0779 


218 1432 


235 1421 


252 0694 


9198 


24 




37 


8230 


167 0556 


184 2373 


3629 


4271 


4248 


3508 


269 2000 


23 




38 


150 1106 


3423 


5232 


6478 


7110 


7075 


6323 


4801 


22 




39 


3981 


6291 


8091 


9327 


9948 


9902 


9137 


7602 


21 




40 


6857 


9159 


185 0949 


202 2176 


219 2786 


236 2729 


253 1952 


270 0403 


20 




41 


9733 


168 2026 


3808 


5024 


5624 


5555 


4766 


3204 


19 




42 


151 2608 


4894 


6666 


7873 


8462 


8381 


7579 


6004 


18 




43 


5484 


7761 


9524 


203 0721 


220 1300 


237 1207 


254 0393 


8305 


17 




44 


8359 


169 0628 


186 2382 


3569 


4137 


4033 


3206 


271 1605 


16 




45 


152 1234 


3495 


5240 


6418 


6974 


6859 


6019 


4404 


15 




46 


4109 


6362 


8098 


9265 


9811 


9684 


8832 


7204 


14 




47 


6984 


9228 


187 0956 


204 2113 


221 2648 


238 2510 


255 1645 


272 0003 


13 




48 


9858 


170 2095 


3813 


4961 


5485 


5335 


4458 


2802 


12 




49 


153 2733 


4961 


6670 


7808 


8321 


8159 


7270 


5601 


11 




50 


5607 


782S 


9528 


205 0655 


222 1158 


239 0984 


256 0082 


8400 


10 




51 


8482 


171 0694 


188 2385 


3502 


3994 


3808 


2894 


273 119S 


9 




52 


154 1356 


3560 


5241 


6349 


6830 


6633 


5705 


3997 


8 




53 


4230 


6425 


8098 


9195 


9666 


9457 


8517 


6794 


7 




54 


7104 


9291 


189 0954 


206 2042 


223 2501 


240 2280 


257 1328 


9595 


6 




55 


9978 


172 2156 


3811 


4888 


5337 


5104 


4139 


274 239C 


5 




56 


155 2851 


5022 


6667 


7734 


8172 


7927 


6950 


5187 


4 




57 


5725 


7887 


9523 


207 0580 


224 1007 


241 0751 


976C 


798^ 


3 




58 


8598 


173 0752 


190 2379 


3426 


3842 


3574 


258 257C 


275 078] 


2 




59 


156 1472 


3617 


5234 


6272 


6676 


6396 5381 


3577 


1 




60 


4345 


6482 


809C 


9117 


9511 


9219 819C 


637^ 







' 


810 


SQo 


79° ] 78° 


77° 


76° 75° 


74° 


' 




NAT. COSINE. 





Table iii.] nat. tan. 165 || 


' 


8° 


[ 9° 


10° 


11° 


120 


13° 


14° 


15° 


/ 





140 5408 


158 3844 


176 3270 


194 3803 


212 5566 


230 8682 


249 3280 


267 9492 


60 


1 


8375 


6826 


6269 


6822 


8606 


231 1746 


6370 


268 2610 


59 


2 


141 1342 


9809 


9269 


9841 


213 1647 


4811 


9460 


5728 


58 


3 


4308 


159 2791 


177 2269 


195 2861 


4688 


7876 


250 2551 


8847 


57 


, 4 


7276 


5774 


5270 


5881 


7730 


232 0941 


5642 


269 1967 


56 


5 


142 0243 


8757 


8270 


8901 


214 0772 


4007 


8734 


5087 


55 


6 


3211 


160 1740 


178 1271 


196 1922 


3814 


7073 


251 1826 


8207 


54 


7 


6179 


4724 


4273 


4943 


6857 


233 0140 


4919 


270 1328 


53 


8 


9147 


7708 


7274 


7964 


9900 


3207 


8012 


4449 


52 


9 


143 2115 


161 0692 


179 0276 


197 0986 


215 2944 


6274 


252 1106 


7571 


51 


10 


5084 


3677 


3279 


4008 


5988 


9342 


4200 


271 0694 


50 


11 


8053 


6662 


6281 


7031 


9032 


234 2410 


7294 


3817 


49 


12 


144 1022 


9647 


9284 


198 0053 


216 2077 


5479 


253 0389 


6940 


48 ; 


13 


3991 


162 2632 


180 2287 


3076 


5122 


8548 


3484 


272 0064 


47 


14 


6961 


5618 


5291 


6100 


8167 


235 1617 


6580 


3188 


46 


15 


9931 


8603 


8295 


9124 


217 1213 


4687 


9676 


6313 


45 


16 


145 2901 


163 1590 


181 1299 


199 2148 


4259 


7758 


254 2773 


9438 


44 


17 


5872 


4576 


4303 


5172 


7306 


236 0829 


5870 


273 2564 


43 


18 


8842 


7563 


7308 


8197 


218 0353 


3900 


8968 


5690 


42 


19 


146 1813 


164 0550 


182 0313 


200 1222 


3400 


6971 


255 2066 


8817 


41 


20 


4784 


3537 


3319 


4248 


6448 


237 0044 


5165 


274 1945 


40 


21 


7756 


6525 


6324 


7274 


9496 


3116 


8264 


5072 


39 


22 


147 0727 


9513 


9330 


201 0300 


219 2544 


6189 


256 1363 


8201 


38 


23 


3699 


165 2501 


183 2337 


3327 


5593 


9262 


4463 


275 1330 


37 


24 


6672 


5489 


5343 


6354 


8643 


238 2336 


7564 


4459 


36 


25 


9644 


8478 


8350 


9381 


220 1692 


5410 


257 0664 


7589 


35 


26 


148 2617 


166 1467 


184 1358 


202 2409 


4742 


8485 


3766 


276 0719 


34 


27 


5590 


4456 


4365 


5437 


7793 


239 1560 


6868 


3850 


33 


28 


8563 


7446 


7373 


8465 


221 0844 


4635 


9970 


6981 


32 


29 


149 1536 


167 0436 


185 0382 


203 1494 


3895 


7711 


258 3073 


277 0113 


31 


30 


4510 


3426 


3390 


4523 


6947 


240 0788 


6176 


3245 


30 


31 


7484 


6417 


6399 


7552 


9999 


3864 


9280 


6378 


29 


32 


150 0458 


9407 


9409 


204 0582 


222 3051 


6942 


259 23S4 


9512 


28 


33 


3433 


168 2398 


186 2418 


3612 


6104 


241 0019 


5488 


278 2646 


27 


34 


6408 


5390 


5428 


6643 


9157 


3097 


8593 


5780 


26 


35 


9383 


8381 


8439 


9674 


223 2211 


6176 


260 1699 


8915 


25 


36 


151 2358 


169 1373 


187 1449 


205 2705 


5265 


9255 


4805 


279 2050 


24 


37 


5333 


4366 


4460 


5737 


8319 


242 2334 


7911 


5186 


23 


38 


8309 


7358 


7471 


8769 


224 1374 


5414 


261 1018 


8322 


22 


39 


152 1285 


170 0351 


188 0483 


206 1801 


4429 


8494 


4126 


280 1459 


21 


40 


4262 


3344 


3495 


4834 


7485 


243 1575 


7234 


4597 


20 


41 


7238 


6338 


6507 


7867 


225 0541 


4656 


262 0342 


7735 


19 


42 


153 0215 


9331 


9520 


207 0900 


3597 


7737 


3451 


281 0873 


18 


43 


3192 


171 2325 


189 2533 


3934 


6654 


244 0819 


6560 


4012 


17 


44 


6170 


5320 


5546 


6968 


9711 


3902 


9670 


7152 


16 


45 


9147 


8314 


8559 


208 0003 


226 2769 


6984 


263 2780 


282 0292 


15 


46 


154 2125 


172 1309 


190 1573 


3038 


5827 


245 0068 


5891 


3432 


14 


47 


5103 


4304 


4587 


6073 


8885 


3151 


9002 


6573 


13 


48 


8082 


7300 


7602 


9109 


227 1944 


6236 


264 2114 


9715 


12 


49 


155 1061 


173 0296 


191 0617 


209 2145 


5003 


9320 


5226 


283 2857 


11 


50 


4040 


3292 


3632 


5181 


6063 


246 2405 


8339 


5999 


10 


51 


7019 


6288 


6648 


8218 


228 1123 


5491 


265 1452 


9143 


9 


52 


9998 


9285 


9664 


210 1255 


4184 


8577 


4566 


284 2286 


8 


53 


156 2978 


174 2282 


192 2680 


4293 


7244 


247 1663 


7680 


5430 


7 


54 


5958 


5279 


5696 


7331 


229 0306 


4750 


266 0794 


8575 


6 


55 


8939 


8277 


8713 


211 0369 


3367 


7837 


3909 


285 1720 


5 


56 


157 1919 


175 1275 


193 1731 


3407 


6429 


248 0925 


7025 


4866 


4 


57 


4900 


4273 


4748 


6446 


9492 


4013 


267 0141 


8012 


3 


58 


7881 


7272 


7766 


9486 


230 2555 


7102 


3257 


286 1159 


2 


59 


158 0863 


176 0271 


194 0784 


212 2525 


5618 


249 0191 


6374 


4306 


1 


60 


3844 


3270 


3803 


5566 


8682 


3280 


9492 


7454 





' 


81° 


80O 


79° 


78° 


77° 


76° 


75° 


74° 


' 


NAT. COTAN. _J| 



166 NAT. SINE. [TaJZeiii. ll 


/ 


16° 


170 


18° 


19° 


20° 


21° 


22° 


23° 


f 





275 6374 


292 3717 


309 0170 


325 5682 


342 0201 


358 3679 


374 6066 


390 7311 


60 


1 


9170 


6499 


2936 


8432 


2935 


6395 


8763 


9989 


59 


2 


276 1965 


9260 


5702 


326 1182 


5668 


9110 


375 1459 


391 2666 


58 


3 


4761 


293 2061 


8468 


3932 


8400 


359 1825 


4156 


5343 


57 


4 


7556 


4842 


310 1234 


6681 


343 1133 


4540 


6852 


8019 


56 


5 


277 0352 


7623 


3999 


9430 


3865 


7254 


9547 


392 0695 


55 


6 


3147 


294 0403 


6764 


327 2179 


6597 


9968 


376 2243 


3371 


54 


7 


5941 


3183 


9529 


4928 


9329 


360 2682 


4938 


6047 


53 


8 


8736 


5963 


311 2294 


7676 


344 2060 


5395 


7632 


8722 


52 


9 


278 1530 


8743 


5058 


328 0424 


4791 


8108 


377 0327 


393 1397 


51 


10 


4324 


295 1522 


7822 


3172 


7521 


361 0821 


3021 


4071 


50 


11 


7118 


4302 


312 0586 


5919 


345 0252 


3534 


5714 


6745 


49 


12 


9911 


7081 


3349 


8666 


2982 


6246 


8408 


9419 


48 


13 


279 2704 


9859 


6112 


329 1413 


5712 


8958 


378 1101 


394 2093 


47 


14 


5497 


296 2638 


8875 


4160 


8441 


362 1669 


■ 3794 


4766 


46 


15 


8290 


5416 


313 1638 


6906 


346 1171 


4380 


6486 


7439 


45 


16 


280 1083 


8194 


4400 


9653 


3900 


7C91 


9178 


395 0111 


44 


17 


3875 


297 0971 


7163 


320 2398 


6628 


9802 


379 1870 


2783 


43 


18 


6667 


3749 


9925 


5144 


9357 


363 2512 


4562 


5455 


42 


19 


9459 


6526 


314 2686 


7889 


347 2085 


5222 


7253 


8127 


41 


20 


281 2251 


9303 


5448 


331 0634 


4812 


7932 


9944 


396 0798 


40 


21 


5042 


298 2079 


8209 


3379 


7540 


364 0641 


380 2634 


3468 


39 


22 


7833 


4856 


315 0969 


6123 


348 0267 


3351 


5324 


6139 


38 


23 


282 0624 


7632 


3730 


8867 


2994 


6059 


8014 


8809 


37 


24 


3415 


299 0408 


6490 


332 1611 


5720 


8768 


381 0704 


397 1479 


36 


25 


6205 


3184 


9250 


4355 


8447 


365 1476 


3393 


4148 


35 


26 


8995 


5959 


316 2010 


7098 


349 1173 


4184 


6082 


6818 


34 


.27 


283 1785 


8734 


4770 


9841 


3898 


6891 


8770 


9486 


33 


28 


4575 


300 1509 


7529 


333 2584 


6624 


9599 


382 1459 


398 2155 


32 


29 


7364 


4284 


317 0288 


5326 


9349 


366 2306 


4147 


4823 


31 


30 


284 0153 


7058 


3047 


8069 


350 2074 


5012 


6834 


7491 


30 


31 


2942 


9832 


5805 


334 0810 


4798 


7719 


9522 


399 0158 


29 


32 


5731 


301 2606 


8563 


3552 


7523 


367 0425 


383 2209 


2825 


28 


33 


8520 


5380 


318 1321 


6293 


351 0246 


3130 


4895 


5492 


27 


34 


285 1308 


8153 


4079 


9034 


2970 


5836 


7582 


8158 


26 


35 


4096 


302 0926 


6836 


335 1775 


5963 


8541 


384 0268 


400 0825 


25 


36 


6884 


3699 


9593 


4516 


8416 


368 1246 


2953 


3490 


24 


37 


9671 


6471 


319 2350 


7256 


352 1139 


3950 


5639 


6156 


23 


38 


286 2458 


9244 


5106 


9996 


3862 


6654 


8324 


8821 


22 


39 


5246 


303 2016 


7863 


336 2735 


6584 


9358 


385 1008 


401 1486 


21 


40 


8032 


47S8 


320 0619 


5475 


9306 


369 2061 


3693 


4150 


20 


41 


287 0819 


7559 


3374 


8214 


353 2027 


4765 


6377 


6814 


i9 


42 


3605 


304 0331 


6130 


337 0953 


4748 


7468 


9060 


9478 


18 


43 


6391 


3102 


8885 


3691 


7469 


370 0170 


386 1744 


402 2141 


17 


44 


9177 


5872 


321 1640 


6429 


354 0190 


2872 


4427 


4804 


16 


45 


288 1963 


8643 


4395 


9167 


2910 


5574 


7110 


7467 


15 


46 


4748 


305 1413 


7149 


338 1905 


5630 


8276 


9792 


403 0129 


14 


47 


7533 


4183 


9903 


4642 


8350 


371 0977 


387 2474 


2791 


13 


48 


289 0318 


6953 


322 2657 


7379 


355 1070 


3678 


5156 


5453 


12 


49 


3103 


9723 


5411 


339 0116 


3789 


6379 


7837 


8114 


11 


50 


5887 


306 2492 


8164 


2852 


6508 


9079 


388 0518 


404 0775 


10 


51 


8671 


5261 


323 0917 


5589 


9226 


372 1780 


3199 


3436 


9 


52 


390 1455 


8030 


3670 


8325 


356 1944 


4479 


5880 


6096 


8 


53 


4239 


307 0798 


6422 


340 1060 


4562 


7179 


8560 


8756 


7 


54 


7022 


3566 


9174 


3796 


7380 


9878 


389 1240 


405 1416 


6 


55 


9805 


6334 


324 1926 


6531 


357 0097 


373 2577 


3919 


4075 


5 


56 


291 2588 


9102 


4678 


9265 


2814 


5275 


6598 


6734 


4 


57 


5371 


308 1869 


7429 


341 2000 


5531 


7973 


9277 


9393 


3 


58 


8153 


4636 


325 0180 


4734 


8248 


374 0671 


390 1955 


406 2051 


2 


59 


292 0935 


7403 


2931 


7458 


358 0964 


3369 


4633 


4709 


1 


60 


3717 


309 0170 


5682 


342 0201 


3679 


6066 


7311 


7366 





' 


730 


720 


71° 


70° 


69° 


68° 


67° 


m<=> 


' 


NAT. COSINE. 1 



J^Taile 111.] NAT. TAN. 167| 


' 


16° 


170 


180 


19° 


20° 


21° 


22° 


23° 


, 


^ 


286 7454 


305 7307 


324 9197 


344 3276 


363 9702 


383 8640 


404 0262 


424 4748 


60 


11 1 


287 0602 


306 0488 


325 2413 


6530 


364 2997 


384 1978 


3646 


8182 


59 


2 


3751 


3670 


5630 


9785 


6292 


5317 


7031 


425 1616 


58 


3 


6900 


6852 


8848 


345 3040 


9588 


8656 


405 0417 


5051 


57 


4 


288 0050 


307 0034 


326 2066 


6256 


365 2885 


385 1996 


3804 


8487 


56 


5 


3201 


3218 


5284 


9553 


6182 


5337 


7191 


426 1924 


55 


6 


6352 


6402 


8504 


346 2810 


9480 


8679 


406 0579 


5361 


54 


7 


9503 


9586 


327 1724 


6068 


366 2779 


386 2021 


3968 


8800 


53 


8 


289 2655 


308 2771 


4944 


9327 


6079 


5364 


7358 


427 2239 


52 


9 


5808 


5957 


8165 


347 2586 


9379 


8708 


407 0748 


5680 


51 


10 


8961 


9143 


328 1387 


5846 


367 2680 


387 2053 


4139 


9121 


50 


n 


290 2114 


309 2330 


4610 


9107 


5981 


5398 


7531 


428 2563 


49 


12 


5269 


5517 


7833 


348 2369 


9284 


8744 


408 0924 


6005 


48 


13 


8423 


8705 


329 1056 


5630 


368 2587 


3SS 2091 


4318 


9449 


47 


14 


291 1578 


310 1893 


4281 


8893 


5890 


5439 


7713 


429 2894 


46 


15 


4734 


5083 


7505 


349 2156 


9195 


8787 


409 llOS 


6339 


45 


16 


7890 


8272 


330 0731 


5420 


369 2500 


389 2136 


4504 


9785 


44 


17 


292 1047 


311 1462 


3957 


8685 


5806 


5486 


7901 


430 3232 


43 


18 


4205 


4653 


7184 


350 1950 


9112 


8837 


410 1299 


6680 


42 


19 


7363 


7845 


331 0411 


5216 


370 2420 


390 2189 


4697 


431 0129 


41 


20 


293 0521 


312 1036 


3639 


8483 


5728 


5541 


8097 


3579 


40 


21 


3680 


4229 


6868 


351 1750 


9036 


8894 


411 1497 


7030 


39 


22 


6839 


7422 


332 0097 


5018 


371 2346 


391 2247 


4898 


432 0481 


38 


23 


9999 


313 0616 


3327 


8287 


5656 


5602 


8300 


3933 


37 


24 


294 3160 


3810 


6557 


352 1556 


8967 


8957 


412 1703 


7386 


36 


25 


6321 


7005 


9788 


482G 


372 2278 


392 2313 


5106 


433 0840 


35 


26 


9483 


314 0200 


333 3020 


8096 


5590 


5670 


8510 


4295 


34 


27 


295 2645 


3396 


6252 


353 1368 


8903 


9027 


413 1915 


7751 


33 


28 


5808 


6593 


9485 


4640 


373 2217 


393 2386 


5321 


434 1208 


32 


29 


8971 


9790 


334 2719 


7912 


5532 


5745 


8728 


4665 


31 


30 


296 2135 


315 2988 


5953 


354 1186 


8847 


9105 


414 2136 


8124 


30 


31 


5299 


6186 


9188 


4460 


374 2163 


394 2465 


5544 


435 1583 


29 


32 


8464 


9385 


335 2424 


7734 


5479 


5827 


8953 


5043 


28 


33 


297 1630 


316 2585 


5660 


355 1010 


8797 


9189 


415 2363 


8504 


27 


34 


4796 


5735 


8896 


4286 


375 2115 


395 2552 


5774 


436 1966 


26 


35 


7962 


8986 


336 2134 


7562 


5433 


5916 


9186 


5429 


25 


36 


298 1129 


317 2187 


5372 


356 0840 


8753 


9280 


416 2598 


8893 


24 


37 


4297 


5389 


8610 


4118 


376 2073 


396 2645 


6012 


437 2357 


23 


88 


7465 


8591 


337 1850 


7397 


5394 


6011 


9426 


5823 


22 


39 


299 0634 


318 1794 


5090 


357 0676 


8716 


9378 


417 2841 


9289 


21 


40 


3803 


4998 


8330 


3956 


377 2038 


397 2746 


6257 


438 2756 


20 


41 


6973 


8202 


338 1571 


7237 


5361 


6114 


9673 


6224 


19 


42 


300 0144 


319 1407 


4813 


358 0518 


8685 


9483 


418 3091 


9693 


18 


43 


3315 


4613 


8056 


3801 


378 2010 


398 2853 


6509 


439 3163 


17 


.44 


6486 


7819 


339 1299 


7083 


5335 


6224 


9928 


6634 


16 


45 


9658 


320 1025 


4543 


359 0367 


8661 


9595 


419 3348 


440 0105 


15 ' 


46 


301 2831 


4232 


7787 


3651 


379 1988 


399 2968 


6769 


3578 


14 


47 


6004 


7440 


340 1032 


6936 


5315 


6341 


420 0190 


7051 


13 


48 


9178 


321 0649 


4278 


360 0222 


8644 


9715 


3613 


441 0526 


12 


49 


302 2352 


3858 


7524 


3508 


380 1973 


400 3089 


7036 


4001 


11 


50 


5527 


7067 


341 0771 


6795 


5302 


6465 


421 0460 


7477 


10 


51 


8703 


322 0278 


4019 


361 0082 


8633 


9841 


3885 


442 0954 


9 


52 


303 1879 


3489 


7267 


3371 


381 1964 


401 3218 


7311 


4432 


8 


53 


5055 


6700 


342 0516 


6660 


5296 


6596 


422 0738 


7910 


7 


54 


8232 


9912 


3765 


9949 


8629 


9974 


4165 


443 1390 


6 


55 


304 1410 


323 3125 


7015 


362 3240 


382 1962 


402 3354 


7594 


4871 


5 


56 


4588 


6338 


343 0266 


6531 


5296 


6734 


423 1023 


8352 


4 


57 


7767 


9552 


3518 


9823 


8631 


403 0115 


4453 


444 1834 


3 


58 


305 0946 


3242766 


6770 


363 3115 


383 1967 


3496 


7884 


5318 


2 


&9 


4126 


5981 


344 0023 


6408 


5303 


6879 


424 1316 


8802 


1 


60 


7307 


9197 


3276 


9702 


8640 


404 0262 


4748 


445 2287 





/ 


73° 


72° 


710 


70° 


69° 


680 


67° 


660 


/ 


NAT. COT AN. || 



J68 NAT. SINE. [Table ,111. 




1 


24° 


25° 


26° 


27° 


28° 


r 29° 


30° 


31° 1 ' 







406 7366 


422 6183 


438 3711 


453 9905 


469 4716 


484 8096 


500 0000 


515 0381 


60 




1 


407 0024 


8819 


6326 


454 2497 


7284 


485 0640 


2519 


2874 


59 




2 


2681 


423 1455 


8940 


5088 


9852 


3184 


5037 


5367 


58 




3 


5337 


4090 


439 1553 


7679 


470 2419 


5727 


7556 


7859 


57 




4 


7993 


6725 


4166 


455 0269 


4986 


8270 


501 0073 


516 0351 


56 




5 


408 0649 


9360 


6779 


2859 


7553 


486 0812 


2591 


2842 


55 




6 


3305 


424 1994 


9392 


5449 


471 0119 


3354 


5107 


5333 


54 




7 


5960 


4628 


440 2004 


8038 


2685 


5895 


7624 


7824 


53 




8 


8615 


7262 


4615 


456 0627 


5250 


8436 


502 0140 


517 0314 


52 




9 


409 1269 


9895 


7227 


3216 


7815 


487 0977 


2655 


2804 


51 




10 


3923 


425 2528 


9838 


5804 


472 0380 


3517 


5170 


5293 


50 




11 


6577 


5161 


441 2448 


8392 


2944 


6057 


7685 


7782 


49 




12 


9230 


7793 


5059 


457 0979 


5508 


8597 


503 0199 


518 0270 


48 




13 


410 1883 


426 0425 


7668 


3566 


8071 


488 1136 


2713 


2758 


47 




14 


4536 


3056 


442 0278 


6153 


473 0634 


3674 


. 5227 


5246 


46 




15 


7189 


5687 


2887 


8739 


3197 


6212 


7740 


7733 


45 




16 


9841 


8318 


5496 


458 1325 


5759 


8750 


504 0252 


519 0219 


44 




17 


411 2492 


427 0949 


8104 


3910 


8321 


489 1288 


2765 


2705 


43 




18 


5144 


3579 


443 0712 


6496 


474 0882 


3825 


5276 


5191 


42 




19 


7795 


6208 


3319 


9080 


3443 


6361 


7788 


7676 


41 




20 


412 0445 


8838 


5927 


459 1665 


6004 


8897 


505 0298 


520 0161 


40 




21 


3096 


428 1467 


8534 


4248 


8564 


490 1433 


2809 


2646 


39 




22 


5745 


4095 


444 1140 


6832 


475 1124 


3968 


5319 


5130 


38 




23 


8395 


6723 


3746 


9415 


3683 


6503 


7828 


7613 


37 




24 


413 1044 


9351 


6352 


460 1998 


6242 


9038 


506 0338 


521 0096 


36 




25 


3693 


429 1979 


8957 


4580 


8801 


491 1572 


2846 


2579 


35 




26 


6342 


4606 


445 1562 


7162 


476 1359 


4105 


5355 


5061 


34 




27 


8990 


7233 


4167 


9744 


3917 


6638 


7863 


7543 


33 




28 


414 1638 


9859 


6771 


461 2325 


6474 


9171 


507 0370 


522 0024 


32 




29 


4285 


430 2485 


9375 


4906 


9031 


492 1704 


2877 


2505 


31 




30 


6932 


5111 


446 1978 


7486 


477 1588 


4236 


5384 


4986 


30 




31 


9579 


7736 


4581 


462 0066 


4144 


6767 


7890 


7466 


29 




32 


415 2226 


431 0361 


7184 


2646 


6700 


9298 


508 0396 


9945 


28 




33 


4872 


2986 


9786 


5225 


9255 


493 1829 


2901 


523 2424 


27 




34 


7517 


5610 


447 2388 


7804 


478 1810 


4359 


5406 


4903 


26 




35 


416 0163 


8234 


4990 


463 0382 


4364 


6889 


7910 


7381 


25 




36 


2808 


432 0857 


7591 


2960 


6919 


9419 


509 0414 


9859 


24 




37 


5453 


3481 


448 0192 


5538 


9472 


494 1948 


2918 


524 2336 


23 




38 


8097 


6103 


2792 


8115 


479 2026 


4476 


5421 


4813 


22 




39 


417 0741 


8726 


5392 


464 0692 


4579 


7005 


7924 


7290 


21 




40 


3385 


433 1348 


7992 


3269 


7131 


9532 


510 0426 


9766 


20 




41 


6028 


3970 


449 0591 


5845 


9683 


495 2060 


2928 


525 2241 


19 




42 


8671 


6591 


3190 


8420 


480 2235 


4587 


5429 


4717 


18 




43 


418 1313 


9212 


5789 


465 0996 


4786 


7113 


7930 


7191 


17 




44 


3956 


434 1832 


8387 


3571 


7337 


9639 


511 0431 


9665 


16 




45 


6597 


4453 


450 0984 


6145 


9888 


496 2165 


2931 


526 2139 


15 




46 


9239 


7072 


3582 


8719 


481 2438 


4690 


5431 


4613 


14 




47 


419 1880 


9692 


6179 


466 1293 


4987 


7215 


7930 


7085 


13 




48 


4521 


435 2311 


8775 


3866 


7537 


9740 


512 0429 


9558 


12 




49 


7161 


4930 


451 1372 


6439 


482 0086 


497 2264 


2927 


527 2030 


11 




50 


9801 


7548 


3967 


9012 


2634 


4787 


5425 


4502 


10 




51 


420 2441 


436 0166 


6563 


467 1584 


5182 


7310 


7923 


6973 


9 




52 


5080 


2784 


9158 


4156 


7730 


9833 


513 0420 


9443 


8 




53 


7719 


5401 


452 1753 


6727 


483 0277 


498 2355 


2916 


528 1914 


7 




54 


421 0358 


8018 


4347 


9298 


2824 


4877 


5413 


4383 


6 




55 


2996 


437 0634 


6941 


468 1869 


5370 


7399 


7908 


6853 


5 




56 


5634 


3251 


9535 


4439 


7916 


9920 


514 0404 


9322 


4 




57 


8272 


5866 


453 2128 


7009 


484 0462 


499 2441 


2899 


529 1790 


3 




58 


422 0909 


8482 


4721 


9578 


3007 


4961 


5393 


4258 


2 




59 


3546 


438 1097 


7313 


469 2147 


5552 


7481 


7887 


6726 


1 




60 


6183 


3711 


9905 


4716 


8096 


500 0000 


515 0381 


9193 







/ 


65° 


64° 


63° 


62° 


61° 


60° 


59° 


58°. ' 




NAT. COSINE. ) 





^Table in.] nat. tan. 169 I| 


1 / 


24° 


25° 


26° 


27° 


28° 


29° 


30° 


31° 


/ 





445 2287 


466 3077 


487 7326 


509 5254 


531 7094 


554 3091 


577 3503 


600 8606 


60 


1 


5773 


6618 


488 0927 


8919 


532 0826 


6694 


7362 


601 2566 


59! 


2 


9260 


467 0161 


4530 


510 2585 


4559 


555 0698 


578 1262 


6527 


58 


3 


446 2747 


3705 


8133 


6252 


8293 


4504 


5144 


602 0490 


57 


4 


6236 


7250 


489 1737 


9919 


533 2029 


8311 


9027 


4454 


56 


5 


9726 


468 0796 


5343 


511 3588 


5765 


556 2119 


579 2912 


6419 


55 


6 


447 3216 


4342 


8949 


7259 


9503 


5929 


6797 


603 2386 


54 


7 


6708 


7890 


490 2557 


512 0930 


534 3242 


9739 


580 0684 


6354 


53 


8 


448 0200 


469 1439 


6166 


4602 


6961 


557 3551 


4573 


604 0323 


52 


9 


3693 


4988 


9775 


8275 


535 0723 


7364 


8462 


4294 


51 


10 


7187 


8539 


491 3386 


513 1950 


4465 


558 1179 


581 2353 


6266 


50 


11 


449 0682 


470 2090 


6997 


5625 


6208 


4994 


6245 


605 2240 


49 


12 


4178 


5643 


492 0610 


9302 


536 1953 


8811 


582 0139 


6215 


48 


13 


7675 


9196 


4224 


514 2980 


5699 


559 2629 


4034 


606 0192 


47 


14 


450 1173 


471 2751 


7838 


6656 


9446 


6449 


7930 


4170 


46 


15 


4672 


6306 


493 1454 


515 0338 


537 3194 


560 0269 


583 1828 


8149 


45 


16 


8171 


9863 


5071 


4019 


6943 


4091 


5726 


607 2130 


44 


17 


451 1672 


472 3420 


8689 


7702 


536 0694 


7914 


9627 


6112 


43 


18 


5173 


6978 


494 2308 


516 1385 


4445 


561 1736 


584 3528 


608 0095 


42 


19 


8676 


473 0538 


5928 


5069 


8198 


5564 


7431 


4060 


41 


20 


452 2179 


4098 


9549 


8755 


539 1952 


9391 


565 1335 


8067 


40 


21 


5683 


7659 


495 3171 


517 2441 


5707 


562 3219 


5241 


609 2054 


39 


22 


9188 


474 1222 


6794 


6129 


9464 


7048 


9148 


6043 


38 


23 


453 2694 


4785 


496 0418 


9618 


540 3221 


563 0679 


586 3056 


610 0034 


37 


24 


6201 


8349 


4043 


518 3508 


6960 


4710 


6965 


4026 


36 


25 


9709 


475 1914 


7669 


7199 


541 0740 


8543 


587 0676 


8019 


35 


26 


454 3218 


5481 


497 1297 


519 0891 


4501 


564 2378 


4788 


611 2014 


34 


27 


6728 


9048 


4925 


4584 


8263 


6213 


8702 


6011 


33 


28 


455 0238 


476 2616 


8554 


8278 


542 2027 


565 0050 


588 2616 


612 0008 


32 


29 


3750 


6185 


498 2185 


520 1974 


5791 


3888 


6533 


4007 


31 


30 


7263 


9755 


5816 


5671 


9557 


7728 


589 0450 


8006 


30 


31 


456 0776 


477 3326 


9449 


9368 


543 3324 


566 1568 


4369 


613 2010 


29 


32 


4290 


6899 


499 3082 


521 3067 


7092 


5410 


8289 


6013 


28 


33 


7806 


478 0472 


6717 


6767 


544 0862 


9254 


590 2211 


614 0016 


27 


34 


457 1322 


4046 


500 0352 


522 0468 


4632 


567 3098 


6134 


4024 


26 


35 


4839 


7621 


3989 


4170 


8404 


6944 


591 0058 


8032 


25 1 


36 


8357 


4791197 


7627 


7874 


545 2177 


568 0791 


3984 


615 2041 


24 1 


37 


458 1877 


4774 


501 1266 


523 1578 


5951 


4639 


7910 


6052 


23 


38 


5397 


8352 


4906 


5284 


9727 


8466 


592 1839 


616 0064 


22 


39 


8918 


480 1932 


6547 


8990 


546 3503 


569 2339 


5768 


4077 


21 


40 


459 2439 


5512 


502 2189 


524 2698 


7281 


6191 


9699 


8092 


20 


41 


5962 


9093 


5832 


6407 


547 1060 


570 0045 


593 3632 


617 2108 


19 


42 


9486 


481 2675 


9476 


525 0117 


4640 


3699 


7565 


6126 


18 


43 


460 3011 


6258 


503 3121 


3829 


8621 


7755 


594 1501 


618 0145 


17 


44 


6537 


9842 


6768 


7541 


548 2404 


571 1612 


5437 


4166 


16 


45 


461 0063 


482 3427 


504 0415 


526 1255 


6188 


5471 


9375 


6188 


15 


46 


3591 


7014 


4063 


4969 


9973 


9331 


595 3314 


619 2211 


14 


47 


7119 


483 0601 


7713 


8685 


549 3759 


572 3192 


7255 


6236 


13 


48 


462 0649 


4189 


505 1363 


527 2402 


7547 


7054 


596 1196 


620 0263 


12 


49 


4179 


7778 


5015 


6120 


550 1335 


573 0916 


6140 


4291 


11 


50 


7710 


484 1368 


8668 


9639 


5125 


4783 


9064 


8320 


10 


51 


463 1243 


4959 


506 2322 


528 3560 


8916 


8649 


597 3030 


621 2351 


9 


52 


4776 


8552 


5977 


7281 


551 2708 


574 2516 


6976 


6363 


8 ! 


53 


8310 


485 2145 


9633 


529 1004 


6502 


6385 


598 0926 


622 0417 


7 i 


54 


464 1845 


5739 


507 3290 


4727 


552 0297 


575 0255 


4877 


4452 


6 


55 


5382 


9334 


6948 


8452 


4093 


4126 


8826 


8488 


5 


56 


8919 


486 2931 


508 0607 


530 2178 


7890 


7999 


599 2781 


623 2527 


4 


57 


465 2457 


6528 


4267 


5906 


553 1666 


576 1673 


6735 


6566 


3 


58 


5996 


487 0126 


7929 


9634 


5488 


5746 


600 0691 


624 0607 


2i 


59 


9536 


3726 


509 1591 


531 3364 


9288 


9625 


4646 


4650 


1 


60 


466 3077 


7326 


5254 


7094 


554 3091 


577 3503 


8606 


8694 





' 


650 


64^ 


630 , 


62° 


61° 


60° 


59° 


58° 


/ 


NAT. COT AN. il 



15 



170 NAT. SINE. [TaWeni.|j 


/ 


32° 


33° 


34° 


35° 


36° 


37° 


38° 


39° 


' 





529 9193 


544 6390 


559 1929 


573 5764 


587 7853 


601 8150 


615 6615 


629 3204 


60 


1 


530 1659 


8830 


4340 


8147 


583 0206 


602 0473 


3907 


5464 


59 


2 


4125 


545 1269 


675i: 


574 0529 


2558 


2795! 


616 1198 


7724 


58 


3 


6591 


3707 


91621 


2911 


4910 


5117 


3489 


9983 


57 


4 


9057 


6145 


560 1572 
3981 


5292 


7262 


7439 


5780 


630 2242 


56 


5 


531 1521 


8583 


7672 


9613 


9760 


8069 


4500 


55 


6 


3986 


546 1020 


6390 


575 0053 


589 1964 


603 2080 


617 0359 


6758 


54 


7 


6450 


3456 


8798 


2432 


4314 


4400 


2646 


9015 


53 


8 


8913 


5892 


561 1206 


4811 


6663 


6719 


4936 


631 1272 


52 ! 


9 


532 1376 


8328 


3614 


7190 


9012 


9038 


7224 


3528 


51 i 


!10 


3839 


547 0763 


6021 


9568 


590 1361 


604 1356 


9511 


5784 


50 1 


11 


6301 


3198 


8423 


576 1946 


3709 


3674; 


618 1798 


80391 49 II 


12 


8763 


5632 


562 0334 


4323 


6057 


5991 


4034 


632 0293 


43 


13 


533 1224 


8066 


3239 


6700 


8404 


8308 


6370 


2547 


47 


14 


3685 


548 0499 


5645 


9076 


591 0750 


605 0624 


8655 


4800 


46 


15 


6145 


2932 


8049 


577 1452 


3096 


2940 


619 0939 


7053 


45 i 


16 


8605 


5365 


563 04531 


3327 


5442 


5255| 


3224 


9306 


44 1 


17 


534 1065 


7797 


28571 


6202 


778? 


7570| 


5507 


633 1557 


43 ! 


18 


3523 


549 0223 


5260; 


8576 


592 0132 


9334^ 


779C 


3809 


42 


19 


5982 


2659 


7663 


573 0950 


2476 


606 2193 


620 0073 


6059 


41 1 


20 


8440 


5090 


564 0066 


3323 


4819 


4511 


2355 


8310 


40 j 


21 


535 0898 


7520 


2467 


5696 


7163 


6824 


463C 


534 05591 39 ! 


22 


3355 


9950 


4869 


8069 


9505 


9136 


6917 


2306J 38 ! 


23 


5812 


550 2379 


7270, 


579 0440 


593 1847 


607 1447 


9196 


50571 37 i 


24 


8268 


4807 


9670 


2312 


4189 


3753 


621 1476 


7305 36 


25 


536 0724 


7236 


565 2070 


5183 
7553 


6530 


6069 


3757 


9553 35 


26 


3179 


9663 


4469 


8371 


8379 


6C36 


635 1500 34 1 


27 


5634 


551 2091 


6368 


9923 


594 1211 


608 0639 


8314 


• 4046 j 33 i 


28 


8089 


4516 


9267: 


580 2292 


3550 


2998 


622 0592 


6292 32 i 


29 


537 0543 


6944 


566 1665 


4661 


5889 


5306 


2870 


8537 31 


30 


2996 


9370 


4062 


7030 


8228 


7614 


5146 


636 0782 30 


31 


5449 


552 1795 


6459 


9397 


595 0566 


9922 


7423 


30261 29 


32 


7902 


4220 


8356 


581 1765 


2904 


609 2229 


9696 


5270; 28 


33 


538 0354 


6645 


567 1252 


4132 


5241 


4535 


623 1974 


7513! 27 


34 


2806 


9069 


3648 


6498 


7577 


6841 


4248 


9756! 26 


35 


5257 


553 1492 


6043 


6364 


9913 


9147 


6522 


637 199S' 25 


36 


7708 


3915 


8437 


582 1230 


596 2249 


610 1452 


8796 


4240: 24 


37 


539 015S 


6333 


568 0832 


3595 


4584 


3756 


624 1069 


6481 i 23 


38 


2608 


8760 


3225 


5959 


6918 


6060 


3342 


8721 


22 


39 


5058 


554 1132 


5619 


8323 


9252 


8363 


5614 


638 0961 


21 1 


40 


7507 


3603 


8011 


583 0687 


597 1586 


611 0666 


7885 


3201 


20 1 


41 


9955 


6024 


569 0403 


3050 


3919 


2969 


625 0156 


5440 1 19 ' 


42 


540 2403 


8444 


2795 


5412 


6251 


5270 


2427 


7678 i 18 


43 


4S51 


555 0864 


5187 


7774 


8583 


7572 


4696 


9916! 17 


44 


729S 


3283 


7577 


584 0136 


598 0915 


9373 


6966 


639 2153; 16 


45 


9745 


5702 


9968 


2497 


3246 


612 2173 


9235 


4390 


1 15 


46 


541 2191 


8121 


570 2357 


4857 


5577 


4473 


626 1503 


6626 


14 


47 


4637 


556 0539 


4747 


7217 


7906 


6772 


3771 


8862 


13 


48 


7062 


2956 


7136 


9577 


599 0236 


9071 


6035 


640 1097 


12 


49 


9527 


5373 


9524 


585 1936 


2565 


613 1369 


8305 


33321 11 11 


50 


542 1971 


7790 


571 1912 


4294 


4893 


3666 


627 0571 


55661 10 11 


51 


4415 


557 0206 


4299 


6652 


7221 


5964 


2337 


7799 


9 


52 


6859 


2621 


6686 


9010 


1 9549 


8260 


5102 


641 0032 


8 


53 


9302 


5036 


9073 


586 1367 


:600 1876 


614 0556 


7366 


2264 


7 


54 


543 1744 


7451 


572 1459 


3724 


4202 


2852 


9631 


4496 


6 


55 


4187 


9865 


3844 


608C 


6528 


5147 


628 1894 


6726 


5 


56 


6628 


558 2279 


6229 


8435 


8854 


7442 


4157 


8958 


4 


57 


9069 


4692 


6614 


587 079C 


601 1179 


9736 


6420 


642 1189 


3 


58 


544 1510 


7105 


573 0996 


314c 


3503 


615 2029 


8682 


3416 


2 


59 


3951 


9517 


3381 


549c 


) 5827 


4322 


629 0943 


5647 


1 


60 


639C 


559 1929 


5764 


785: 


I 8150 


661£ 


3204 


7876 





/ 


57° 


56° 


55° 


54° 


53° 


52° 


1 51° 


50° 


' 


i__ NAT. COSIKE. t| 



Table iii.] nat. tan 171 || 


/ 


1 S20 


33° 


340 


( 350 


36° 


37° 


38° 


39° 


/ 





624 8694 


649 4076 


674 5085 


700 2075 


726 5425 


753 5541 


781 2856 


809 7840 


60 


1 


625 2739 


8212 


9318 


6411 


9871 


754 0102 


7542 


810 2658 


59 


2 


6786 


650 2350 


675 3553 


701 0749 


727 4318 


4666 


782 2229 


7478 


58 


3 


626 0834 


6490 


7790 


5089 


8767 


9232 


6919 


811 2300 


57 


4 


4884 


651 0631 


676 2028 


9430 


728 3218 


755 3799 


7831611 


7124 


56 


5 


8935 


4774 


6268 


702 3773 


7671 


8369 


6305 


812 1951 


55 


6 


627 2988 


8918 


677 0509 


8118 


729 2125 


756 2941 


784 1002 


6780 


54 


7 


7042 


652 3064 


4752 


703 2464 


6582 


7514 


5700 


813 1611 


53 


8 


628 1098 


7211 


8997 


6813 


730 1041 


757 2090 


785 0400 


6444 


52 


9 


5155 


653 1360 


678 3243 


704 1163 


5501 


6668 


5103 


814 1280 


51 


10 


9214 


5511 


7492 


5515 


9963 


758 1248 


9808 


6118 


50 


11 


629 3274 


9663 


679 1741 


9869 


731 4428 


5829 


786 4515 


815 0958 


49 


12 


7336 


654 3817 


5993 


705 4224 


8894 


759 0413 


9224 


5801 


48 


13 


630 1399 


7972 


680 0246 


8581 


732 3362 


4999 


787 3935 


816 0646 


47 


14 


5464 


655 2129 


4501 


706 2940 


7832 


9587 


8649 


5493 


46 


15 


9530 


6287 


8758 


7301 


733 2303 


760 4177 


788 3364 


817 0343 


45 


16 


631 3598 


656 0447 


681 3016 


707 1664 


6777 


8769 


8082 


5195 


44 


17 


7667 


4609 


7276 


6028 


734 1253 


761 3363 


789 2802 


818 0049 


43 


IS 


632 1738 


8772 


682 1537 


708 0395 


5730 


7959 


7524 


4905 


42 


19 


5810 


657 2937 


5801 


4763 


735 0210 


762 2557 


790 2248 


9764 


41 


20 


9883 


7103 


683 0066 


9133 


4691 


7157 


6975 


819 4625 


40 


21 


633 3959 


658 1271 


4333 


709 3504 


9174 


763 1759 


791 1703 


9488 


39 


22 


8035 


5441 


8601 


7878 


736 3660 


6363 


6434 


820 4354 


38 


23 


634 2113 


9612 


684 2871 


710 2253 


8147 


764 0969 


792 1167 


9222 


37 


24 


6193 


659 3785 


7143 


6630 


737 2636 


5577 


5902 


821 4093 


36 


25 


635 0274 


7960 


685 1416 


711 1009 


7127 


765 0188 


793 0640 


8965 


35 


26 


4357 


660 2136 


5692 


5390 


738 1620 


4800 


5379 


822 3840 


34 


27 


8441 


6313 


9969 


9772 


6115 


9414 


794 0121 


8718 


33 


28 


636 2527 


661 0492 


686 4247 


712 4157 


739 0611 


766 4031 


4865 


823 3597 


32 


.29 


6614 


4673 


8528 


8543 


5110 


8649 


9611 


8479 


31 


30 


637 0703 


8856 


687 2810 


713 2931 


9611 


767 3270 


795 4359 


824 3364 


30 


31 


4793 


662 3040 


7093 


7320 


740 4113 


7893 


9110 


8251 


29 


32 


8885 


7225 


688 1379 


714 1712 


8618 


768 2517 


796 3862 


825 3140 


28 


33 


638 2978 


663 1413 


5666 


6106 


741 3124 


7144 


8617 


8031 


27 


34 


7073 


5601 


9955 


715 0501 


7633 


769 1773 


797 3374 


826 2925 


26 


35 


639 1169 


9792 


689 4246 


4898 


742 2143 


6404 


8134 


7821 


25 


36 


5267 


664 3984 


8538 


9297 


6655 


770 1037 


798 2895 


827 2719 


24 


37 


9366 


8178 


690 2832 


716 3698 


743 1170 


5672 


7659 


7620 


23 


38 


640 3467 


665 2373 


7128 


8100 


5686 


771 0309 


799 2425 


828 2523 


22 


39 


7569 


6570 


691 1425 


717 2505 


744 0204 


4948 


7193 


7429 


21 


40 


641 1673 


666 0769 


5725 


6911 


4724 


9589 


800 1963 


829 2337 


20 


41 


5779 


4969 


692 0026 


718 1319 


9246 


772 4233 


6736 


7247 


19 


42 


9886 


9171 


4328 


5729 


745 3770 


8878 


801 1511 


830 2160 


18 


43 


642 3994 


667 3374 


8633 


719 0141 


8296 


773 3526 


6288 


7075 


17 


44 


8105 


7580 


693 2939 


4554 


746 2824 


8176 


802 1067 


831 1992 


16 


45 


643 2216 


668 1786 


7247 


8970 


7354 


774 2827 


5849 


6912 


15 


46 


6329 


5995 


694 1557 


720 3387 


747 1886 


7481 


805 0632 


832 1834 


14 


47 


644 0444 


669 0205 


5868 


7806 


6420 


775 2137 


5418 


6759 


13 


48 


4560 


4417 


695 0181 


721 2227 


748 0956 


6795 


804 0206 


833 1686 


12 


49 


8678 


8630 


4496 


6650 


5494 


776 1455 


4997 


6615 


11 


50 


645 2797 


670 2845 


8813 


722 1075 


749 003? 


6118 


9790 


834 1547 


10 


51 


6918 


7061 


696 3131 


5502 


45/5 


777 0782 


805 4584 


6481 


9 


52 


646 1041 


671 1280 


7451 


9930 


9119 


5448 


9382 


835 1418 


8 


53 


5165 


5500 


697 1773 


723 4361 


750 3665 


778 0117 


806 4181 


6357 


7 


54 


9290 


9721 


6097 


8793 


8212 


4788 


8983 


836 1298 


6 


55 


647 3417 


672 3944 


698 0422 


724 3227 


751 2762 


9460 


807 3787 


6242 


5 


56 


7546 


8169 


4749 


7663 


7314 


779 4135 


8593 


837 1188 


4 


57 


648 1676 


673 2396 


9078 


725 2101 


752 1867 


8812 


808 3401 


6136 


3 


58 


5808 


6624 


699 3409 


6540 


6423 


780 3492 


8212 


338 1087 


2 


59 


9941 


674 0854 


774 J 726 0982 


753 0981 


8173 


809 3025 


6041 


1 


60 


649 4076 50851 


700 20751 5425 


5541 


781 2856 


7840 


339 0996 





/ 


57° 


56° 1 


550 ! 


540 1 


53° 


52° 


51° 


50° 


' 



NAT. COT AN. 



172 








NAT. 


SINE. 






\Tahle 


III. 


f 


40° 


41° 


42° 1 


43° 


44° 


45° 1 46° 1 


47° 


"T 





642 7876 


656 0590 


669 1306 


381 9984 


594 6584 


707 1068 


719 3398 


731 3537 


60 


1 


643 0104 


2785 


3468 


382 2111 


8676 


3124 


5418 


5521 


59 


2 


2332 


4980 


5628 


4237 


395 0767 


5180 


7438 


7503 


58 


3 


4559 


7174 


7789 


6363 


2858 


7236 


9457 


9486 


57 


4 


6785 


9367 


9948 


8489 


4949 


9291 


720 1476 


732 1467 


56 


5 


9011 


657 1560 


670 2108 


383 0613 


7039 


708 1345 


3494 


3449 


55 


6 


644 1236 


3752 


4266 


2738 


9128 


3398 


5511 


5429 


54 


7 


3461 


5944 


6424 


4861 


396 1217 


5451 


7528 


7409 


53 


8 


5685 


8135 


8582 


6984 


3305 


7504 


9544 


9388 


52 


9 


7909 


658 0326 


671 0739 


9107 


5392 


9556 


721 1559 


733 1367 


51 


10 


645 0132 


2516 


2895 


384 1229 


7479 


709 1607 


3574 


3345 


50 


11 


2355 


4706 


5051 


3350 


9565 


3657 


5589 


5322 


49 


12 


4577 


6895 


7206 


5471 


397 1651 


5707 


7602 


7299 


48 


13 


6798 


9083 


9361 


7591 


3736 


7757 


9615 


9275 


47 


14 


9019 


659 1271 


672 1515 


9711 


5821 


9806 


722 1628 


734 1250 


46 


15 


646 1240 


3458 


3668 


585 1830 


7905 


710 1854 


3640 


3225 


45 


16 


3460 


5645 


5821 


3948 


9988 


3901 


5651 


5199 


44 


17 


5679 


7831 


7973 


6066 


398 2071 


5948 


7661 


7173 


43 


18 


7898 


S60 0017 


673 0125 


8184 


4153 


7995 


9671 


9146 


42 


19 


S47 0116 


2202 


2276 


386 0300 


6234 


711 0041 


723 1681 


735 1118 


41 


20 


2334 


4386 


4427 


2416 


8315 


2086 


3690 


3090 


40 


21 


4551 


6570 


6577 


' 4532 


699 0396 


4130 


5698 


5061 


39 


22 


6767 


8754 


8727 


6647 


2476 


6174 


7705 


7032 


38 


23 


8984 


S61 0936 


674 0876 


8761 


4555 


8218 


9712 


9002 


37 


24 


648 1199 


3119 


3024 


387 0875 


6633 


712 0260 


724 1719 


736 0971 


36 


25 


3414 


5300 


5172 


2988 


8711 


2303 


3724 


2940 


35 


26 


5628 


7482 


7319 


5101 


700 0789 


4344 


5729 


4908 


34 


27 


7842 


9682 


9466 


7213 


2866 


6385 


7734 


6875 


33 


28 


649 0056 


662 1842 


675 1612 


9325 


4942 


8426 


9738 


8842 


32 


29 


2268 


4022 


3757 


388 1435 


7018 


713 0465 


725 1741 


737 0608 


31 


30 


4480 


6200 


5902 


3546 


9093 


2504 


3744 


2773 


30 


31 


6692 


8379 


8046 


5655 


701 1167 


4543 


5746 


4738 


29 


32 


8903 


663 0557 


676 0190 


7765 


3241 


6581 


7747 


6703 


28 


33 


650 1114 


2734 


2333 


9873 


5314 


8618 


9748 


8666 


27 


34 


3324 


4910 


4476 


689 1981 


7387 


714 0655 


726 1748 


738 0629 


26 


35 


5533 


7087 


6618 


4089 


9459 


2691 


3748 


2592 


25 


36 


7742 


9262 


8760 


6195 


702 1531 


4727 


5747 


4553 


24 


37 


9951 


664 1437 


677 0901 


8302 


3601 


6762 


7745 


6515 


23 


38 


651 2158 


3612 


3041 


690 0407 


5672 


8796 


9743 


6475 


22 


39 


4366 


5785 


5181 


2512 


7741 


715 0830 


727 1740 


739 0435' 21 1 


40 


6572 


7959 


7320 


4617 


9811 


2863 


3736 


2394. 20 1 


41 


8778 


665 0131 


9459 


6721 


703 1879 


4895 


5732 


4353 


19 


42 


652 0984 


2304 


678 1597 


8824 


3947 


6927 


7728 


6311 


18 


43 


3189 


4475 


3734 


691 0927 


6014 


8959 


9722 


8268 


17 


44 


5394 


6646 


5871 


3029 


8081 


716 0989 


728 1716 


740 0225 


16 


45 


7598 


8817 


8007 


5131 


704 0147 


3019 


3710 


2181 


15 


46 


9801 


666 0987 


679 0143 


7232 


2213 


5049 


5703 


4137 


14 


47 


653 2004 


3156 


2278 


9332 


4278 


7078 


7695 


6092 


13 


48 


4206 


5325 


4413 


692 1432 


6342 


9106 


9686 


8046 


12 


49 


6403 


7493 


6547 


3531 


8406 


717 1134 


729 1677 


741 0000 


11 


50 


8609 


9661 


8691 


5630 


705 0469 


3161 


3668 


1953 


10 


51 


654 0810 


667 1828 


680 0813 


7728 


2532 


5187 


565^ 


3905 


9 


52 


3010 


3994 


2946 


9825 


4594 


7213 


7646 


5857 


8 


53 


5209 


6160 


5078 


693 1922 


6655 


9238 


9635 


7808 


7 


54 


7408 


8326 


7209 


4018 


8716 


718 1263 


730 1623 


9758 


6 


55 


9607 


668 O490 


9339 


6114 


706 0776 


3287 


3610 


742 1708 


5 


56 


655 1804 


2655 


681 1469 


8209 


2835 


5310 


5597 


3658 


4 


57 


4002 


481£ 


3599 


694 0304 


4894 


7333 


7583 


5606 


3 


58 


6198 


6981 


5728 


2398 


6953 


9355 


9568 


7554 


2 


59 


8395 


9144 


7856 


4491 


9on 


719 1377 


731 1553 


9502 


1 


60 


656 0590 


669 1306 


9984 


6584 


707 1068 3398 


3537 


743 1448 





' 


49° 


48° 


47° 


46° 


45° 44° 


43° 


420 


/ 


NAT. COSINE. ' jl 



Tabic III.] KAT. TA^ 


. 




173 ll 


' 


40O 


41° 


420 


43° 


440 


450 


46° 


470 


' 





8390996 


3692867 


9004040 


9325151 


9656338 


1-0000000 


1-0355303 


1-0723687 


60 


1 


5955 


7976 


9309 


9330591 


9662511 


05819 


61333 


29943 


59 


2 


3400915 


3703037 


9014580 


6034 


8137 


11642 


67367 


36203 


58 


3 


5S7S 


S2C0 


9854 


9341479 


9673767 


17469 


73404 


42467 


57 


4 


8410844 


3713316 


9025131 


6923 


9399 


23298 


79445 


48734 


56 


1 5 


5812 


8435 


9030411 


9352380 


9685035 


29131 


85489 


55006 


55 


6 


84207S2 


8723556 


5693 


7834 


9690674 


34968 


91538 


61282 


54 


7 


5755 


8680 


9040979 


9363292 


6316 


40807 


97569 


67561 


53 


8 


8430730 


8733806 


6267 


8753 


9701962 


46651 


1-0403645 


73845 


52 


9 


5708 


8935 


9051557 


9374216 


7610 


52497 


09704 


80132 


51 


10 


S440688 


8744067 


6851 


9683 


9713262 


58348 


15767 


86423 


50 


11 


5670 


9201 


9062147 


9335153 


8917 


64201 


21633 


92718 


49 


12 


8450655 


8754338 


7446 


9390625 


9724575 


70058 


27904 


99018 


48 


13 


5643 


9478 


9072748 


6101 


9730236 


75918 


33977 


1-0805321 


47 


14 


8460633 


8764620 


8053 


9401579 


5901 


81782 


40055 


11628 


46 


15 


5625 


9765 


9083360 


7061 


9741569 


87649 


46136 


17939 


45 


16 


8470620 


8774912 


8671 


9412545 


7240 


93520 


52221 


24254 


44 


17 


5617 


8780062 


9093984 


8033 


9752914 


99394 


58310 


30573 


43 


18 


8480617 


5215 


9300 


9423523 


8591 


1-0105272 


64402 


36896 


42 


19 


5619 


8790370 


9104919 


9017 


9764272 


11153 


70498 


43223 


41 


20 


8490624 


5528 


9940 


9434513 


9956 


17033 


76598 


49554 


40 


21 


5631 


8800688 


9115265 


9440013 


9775643 


22925 


82702 


55889 


39 


22 


8500640 


5852 


9120592 


5516 


9781333 


28817 


88809 


62228 


38 


23 


5653 


8811017 


5922 


9451021 


7027 


34712 


94920 


68571 


37 


24 


8510667 


6188 


9131255 


6530 


9792724 


40610 


1-0501034 


74918 


36 


25 


5684 


8821357 


6591 


9462042 


8424 


46512 


07153 


81269 


35 


26 


8520704 


6531 


9141929 


7556 


9804127 


52418 


13275 


87624 


34 


27 


5726 


8831707 


7270 


9473074 


9833 


58326 


19401 


93984 


33 


28 


8530750 


6886 


9152615 


8595 


9815543 


64239 


25531 


1-0900347 


32 


29 


5777 


8842068 


7962 


9484119 


9821256 


70155 


31664 


06714 


31 


30 


8540807 


7253 


9163312 


9646 


6973 


76074 


37801 


13085 


30 


31 


5839 


3852440 


8665 


9495176 


9832692 


81997 


43942 


19460 


29 


32 


8550873 


7630 


9174020 


9500709 


8415 


87923 


50087 


25840 


28 


33 


5910 


8362822 


9379 


6245 


9344141 


93853 


56235 


32223 


27 


34 


8560950 


8017 


9184740 


9511784 


9371 


99786 


62383 


38610 


26 


35 


5992 


8873215 


9190104 


7326 


9855603 


1-0205723 


68544 


45002 


25 


36 


8571037 


8415 


5471 


9522871 


9861339 


11664 


74704 


51397 


24 


37 


6084 


8883619 


9200841 


8420 


7079 


17608 


80867 


57797 


23 


38 


8581133 


8825 


6214 


9533971 


9872821 


23555 


87035 


64201 


22 


39 


6185 


8894033 


9211590 


9526 


8567 


29506 


93206 


70609 


21 


40 


8591240 


9244 


6969 


9545083 


9884316 


35461 


99381 


770-20 


20 


41 


6297 


8904458 


9222350 


9550644 


9890069 


41419 


1-0605560 


83436 


19 


42 


8601357 


9675 


7734 


6208 


5825 


47381 


11742 


89857 


18 


43 


6419 


8914894 


9233122 


9561774 


9901584 


53346 


17929 


96281 


17 


44 


8611484 


8920116 


8512 


7344 


7346 


59315 


24119 


1-1002709 


16 


45 


6551 


5341 


9243905 


9572917 


9913112 


65287 


30313 


09141 


15 


46 


8621621 


8930569 


9301 


8494 


8881 


71263 


36511 


15578 


14 


47 


6694 


5799 


9254700 


9584073 


9924654 


77243 


42713 


22019 


13 


48 


8631768 


8941032 


9260102 


9655 


9930429 


83226 


48918 


28463 


12 


49 


6846 


6268 


5506 


9595241 


6208 


89212 


55128 


34912 


11 


50 


8641926 


8951506 


9270914 


9600829 


9941991 


95203 


61341 


41365 


10 


51 


7009 


6747 


6324 


6421 


7777 


1-0301196 


67558 


47823 


9 


52 


8652094 


8961991 


9281738 


9612016 


9953566 


07194 


73779 


54284 


8 


53 


7181 


7238 


7154 


7614 


9358 


13195 


80004 


60750 


7 


54 


8662272 


8972487 


9292573 


9623215 


9965154 


19199 


86233 


67219 


6 


55 


7365 


7739 


7996 


8819 


9970953 


25208 


92466 


73693 


5 


56 


8672460 


8982994 


9303421 


9634427 


6756 


31220 


98702 


80171 


4 


57 


7558 


8251 


8849 


9640037 


9982562 


37235 


1-0704943 


86653 


3 


58 


8682659 


8993512 


9314280 


5651 


8371 


43254 


11187 


93140 


2 


59 


7762 


8775 


9714 


9651268 


9994184 


49277 


17435 


99630 


1 


60 


8692867 


9004040 9325151 


6888 


1-0000000 


55303 


23687 


1-1106125 





' 


49° 


480 47° 


46° 


45° 


440 


43° 


42° 


' 


NAT. COTAN 






\ 



15* 



^ 


NAT. SINE. [Table iii. [ 




'^~ 


48^ 


49^ 


50^ 


51° 1 5-2° f 53° 


54° 


55° 


/ 







743 1448 


754 7096 


766 0444 


777 1460'7&3 0103 798 6355 


809 0170 


319 1520 


60 




1 


3394 


9004 


2314 


3290 


1898 8105 


1879 


3139 


59 




2 


5340 


755 0911 


4183 


5120 


3683 9855 


3583 


4856 


58 




3 


7285 


281S 


6051 


6949 


5477 799 1604 


5296 


6523 


57 




4 


9229 


4724 


7918 


6777 


7266, 3352 


7004 


8189 


56 




5 


744 1173 


6630 


9735 


778 0604: 90541 5100 


8710 


9854 


55 




6 


3115 


8535 


767 1652 


2431739 03411 6347 


810 0416 


320 1519 


54 




7 


5058 


756 0439 


3517 


4258 26271 8593 


2122 


3183 


53 




8 


6999 


2342 


5382 


6034 


4413800 0338 


3826 


4846 


52 




9 


8941 


4246 


7246 


7909 


6198 


2033 


5530 


6509 


51 




10 


745 0881 


6148 


9110 


9733 


7983 


3827 


7234 


8170 


50 




11 


2321 


8050 


768 0973 


779 1557 


9767 


5571 


6936 


9332 


49 




12 


4760 


9951 


2835 


3330 


790 1550 


7314 


311 0638 


821 1492 


48 




13 


6699 


757 1351 


4697 


5202 


3333, 9056 


2339 


3152 


47 




14 


8636 


6751 


6553 


7024 


5115 6010797 


4040 


4811 


46 




15 


746 0574 


5650 


8413 


6845 


6896 


2533 


5740 


8469 


45 




16 


2510 


7543 769 0273| 


780 0665 


8676 


4278 


7439 


8127 


44 




17 


4446 


9446 


2137 


2435 791 0456; 


6013 


9137 


9784 


43 




18 


6382 


758 1343 


3996 


4304 2235! 


7756 312 03351 


322 1440 


42 




19 


8317 


3240 


5353 


6123 


4014 


9495 


2532 


3096 


41 




20 


747 0251 


5136 


7710 


7940 


5792 


802 1232 


4229 


4751 


40 




21 


2184 


7031 


9567 


9757 


7569 


2969 


5925 


6405 


39 




22 


4117 


8926 


770 1423 


781 1574 


9345 


4705 


7620 


8059 


38 




23 


6049 


759 0820 


3273 


3390792 11211 


6440 


9314 


9712 


37 




24 


7931 


2713 


5132 


5205 


2896 


8175 


813 1003 


823 1364 


36 




25 


9912 


4606 


6986 


7019 


4671 


9909 


2701 


3015 


35 




! 26 


748 1642 


6493 


6840 


8833 


6445 


803 1642 


4393 


4666 


34 




,27 


3772 


8389 


771 0692 


762 0646 


82181 3375 


6084 


6316 


33 




i28 


5701 


760 0230 


2544 


2459 i 9990 


5107 


7775 


7965 


32 




29 


7629 


2170 


4395 


4270 793 1762 


6838 


9466 


9614 


31 




|30 


9557 


4060 


6246 


6032 3533 


8569 


614 1155 


324 1262 


30 




31 


749 1484 


5949 


6096 


7892 5304 804 0299 


2344 


2909 


29 




32 


3411 


7837 


9945 


97021 7074 


2028 


4532 


4556 


28 




33 


5337 


9724 


772 1794 


783 15111 8343 


3756 


6220 


6202 


27 




34 


7262 


761 1611 


3642 


3320 794 0611 


5464 


7906 


7847 


26 




35 


9187' 


3497 


5439 


51271 


2379 


7211 


9593 


9491 


25 




36 


750 nil 


5333 


7336 


6935 


4146 


6938 815 12731 


325 1135 


24 




37 


3034 


7263 


9182 


8741 


5913 


805 0664 


2963 


2778 


23 




33 


4957 


9152 773 10271 


734 0547 


7678 


2389 


4647 


4420 


22 




39 


6879 


762 1036 


2372 


2352 


9444 


4113 


6330 


6062 


21 




40 


8800 


2919 


4716 


4157 


795 1206 


5837 


8013 


7703 


20 




41 


751 0721 


4802 


6559 


5961 


2972 


7560 


9695 


9343 


19 




42 


2641 


6633 


6402 


7764 


4735 


9233:816 1376 


326 0983 


18 




43 


4561 


8564 


774 0244 


9566 


6497!506 1005 


3056 


2622 


17 




44 


6480 


763 0445 


2036 


735 1363 


8259 


2726 


4736 


4260 


16 




45 


8398 


2325 


3926 


3169:796 0020 


4446 


6416 


5897 


15 




46 


752 0316 


4204 


5767 


4970 


1780 


6166 


8094 


7534 


14 




47 


2233 


6082 


7606 


6770 


3540 


7835 


9772 


9170 


13 




48 


4149 


7960 


9445 


8569 


5299 


9603 


817 1449 


727 0806 


12 




49 


6065 


9833 


775 1233 


786 0367 


7056 


807 1321 


3125 


2440 


11 




50 


7980 


764 1714 


3121 


2165 


8815 


3033 


4801 


4074 


10 




51 


9894 


3590 


4957 


39631797 0572 


4754 


6476 


5708 


9 




52 


753 1808 


5465 


6794 


• 5759 


2329 


6470 


8151 


7340 


8 




53 


3721 


7340 


8629 


7555 


4034 


6135 


9824 


8972 


7 




54 


5634 


92141776 0464 


9350 


5839 


9S% 


818 1497 


828 0603 


6 




55 


7546 


765 1087 


1 2298 


787 1145 


7594 


808 1615 


3169 


2234 


5 




66 


9457 


296C 


4132 


2939 


9347 


332= 


4841 


3864 


4 




37 


754 136? 


433'. 


5965 


4732 


793 HOC 


503' 


6512 


5493 


3 




58 


327c 


670^ 


7797 


652^ 


2853 


674C 


8162 


7121 


2 




59 


518- 


857^ 


[\ 9629 


8316 


4604 


846C 


) 9852 


874C 


1 




1 60 


im 


) 766 0444,777 1460 


768 OlOE 


635E 


809 017( 


) 819 1520 


829 0376 







1 ' 


41° 


40^ 1 39° 


38° 


37° 


36° 


35<' 


34° 


' 




L 


NAT. COSINE. 





Table in.] 






NAT 


. TAN. 


' 


" 


175]] 


T 


48° 


49° 


50° 


51° 


52° 


530 


540 


55° 


/ 





1-1106125 


1-1503684 


1-1917536 


1-2348972 


1-2799416 


1-3270448 


1-3763819 


1 -4281480 


60 


1 


12624 


10445 


24579 


56319 


1-2807094 


78483 


72242 


90326 


59 


2 


19127 


17210 


31626 


63672 


14776 


86524 


80672 


99178 


58 


3 


25635 


23979 


38679 


71030 


22465 


94571 


89108 


1-4308039 


57 


4 


32146 


30754 


45736 


78393 


30160 


1-3302624 


97551 


16906 


56 


5 


38662 


37532 


52799 


85^62 


37860 


10684 


1-3806001 


25781 


55 


6 


45182 


44316 


59866 


93136 


45566 


18750 


14458 


34664 


54 


7 


51706 


51104 


66938 


1-2400515 


53277 


26822 


22922 


43554 


53 


' 8 


58235 


57896 


74015 


07900 


60995 


34900 


31392 


52451 


52 


9 


64768 


64693 


81097 


15290 


68718 


42984 


39869 


61356 


51 


10 


71305 


71495 


88184 


22685 


76447 


51075 


48353 


70268 


50 


11 


77846 


78301 


95276 


30086 


84182 


59172 


56844 


79187 


49 


;12 


84391 


85112 


1-2002373 


37492 


91922 


67276 


65342 


88114 


48 


13 


90941 


91927 


09475 


44903 


99669 


75386 


73847 


97049 


47 


14 


97495 


98747 


16581 


52320 


1-2907421 


83502 


82358 


1-4405991 


46 


15 


11204053 


1-1605571 


23693 


59742 


15179 


91624 


90876 


14940 


45 


16 


10616 


12400 


30810 


67169 


22943 


99753 


99401 


23897 


44 


17 


17183 


19234 


37932 


74602 


30713 


1-3407888 


1-3907934 


32862 


43 


18 


23754 


26073 


45058 


82040 


38488 


16029 


16473 


41834 


42 


19 


30329 


32916 


52190 


89484 


46270 


24177 


25019 


50814 


41 


20 


36909 


39763 


59327 


96933 


54057 


32331 


33571 


59801 


40 


21 


43493 


46615 


66468 


1-2504388 


61850 


40492 


42131 


68796 


39 


22 


50081 


53472 


73615 


11848 


69649 


48658 


50698 


77798 


38 


23 


56674 


60334 


80767 


19313 


77454 


56832 


59272 


86808 


37 


24 


63271 


67200 


, 87924 


26784 


85265 


65011 


67852 


95825 


36 


25 


69872 


74071 


95085 


34260 


93081 


. 73198 


76440 


1-4504850 


35 


26 


76478 


80947 


1-2102252 


41742 


1-3000904 


81390 


85034 


13883 


34 


27 


83088 


87827 


09424 


49229 


08733 


89589 


93636 


22923 


33 


28 


89702 


94712 


16601 


56721 


16567 


97794 


1-4002245 


31971 


32 


29 


96321 


1-1701601 


23783 


64219 


24407 


1-3506006 


10860 


41027 


31 


30 


11302944 


08496 


30970 


71723 


32254 


14224 


19483 


50090 


30 


31 


09571 


15395 


38162 


79232 


40106 


22449 


28113 


59161 


29 


32 


16203 


22298 


45359 


86747 


47964 


30680 


36749 


68240 


28 


33 


22839 


29207 


52562 


94267 


55828 


38918 


45393 


77326 


27 


34 


29479 


36120 


59769 


1-2601792 


63699 


47162 


54044 


86420 


26 


35 


36124 


43038 


66982 


09323 


71575 


55413 


62702 


95522 


25 


36 


42773 


49960 


74199 


16860 


79457 


63670 


71367 


1-4604632 


24 


37 


49427 


56888 


81422 


24402 


87345 


71934 


80039 


13749 


23 


38 


56085 


63820 


88650 


31950 


95239 


80204 


88718 


22874 


22 


39 


62747 


70756 


95883 


39503 


1-3103140 


88481 


97405 


32007 


21 


40 


69414 


77698 


1-2203121 


47062 


11046 


96764 


1-4106098 


41147 


20 


41 


76086 


84644 


10364 


54626 


18958 


1-3605054 


14799 


50296 


19' 


42 


82761 


91595 


17613 


62196 


26876 


13350 


23506 


59452 


I81 


43 


89441 


98551 


24866 


69772 


34801 


21653 


32221 


68616 


17 


44 


96126 


1-1805512 


32125 


77353 


42731 


29963 


40943 


77788 


16 


45 


1-1402815 


12477 


39389 


84940 


50668 


38279 


49673 


86967 


15 


46 


09508 


19447 


46658 


92532 


58610 


46602 


58409 


96155 


14 


47 


16206 


26422 


53932 


1-2700130 


66559 


54931 


67153 


1-4705350 


13 


48 


22908 


33402 


61211 


07733 


74513 


63267 


75904 


14553 


12 


49 


29615 


40387 


68496 


15342 


82474 


71610 


84662 


23764 


11 


50 


36326 


47376 


75786 


22957 


90441 


79959 


93427 


32983 


10 


51 


43041 


54370 


83081 


30578 


98414 


88315 


1-4202200 


42210 


9 


52 


49762 


61369 


90381 


38204 


1-3206393 


96678 


10979 


51445 


8 


53 


56486 


68373 


97687 


45835 


14379 


1-3705047 


19766 


60688 


7 


54 


63215 


75382 


1-2304997 


53473 


22370 


13423 


28561 


69938 


6 


55 


69949 


82395 


12313 


61116 


30368 


21806 


37362 


79197 


5 


56 


76687 


89414 


19634 


68765 


36371 


30195 


46171 


88463 


4 


57 


83429 


96437 


26961 


76419 


46381 


38591 


54988 


97738 


3 


58 


90176 


1-1903465 


34292 


84079 


54397 


46994 


63811 


1-4807021 


2 


59 


96928 


10498 


41629 


91745 


62420 


55403 


72642 


16311 


1 


60 


1-1503684 


17536 


48972 


99416 


70448 


63819 


81480 


25610 





/ 


410 


40<^ 


390 


38° 


370 


36° 


35« 


340 


' 










NAT. C 


OTAN. 




15^5=== 


'» 



176 


NAT. SINE. YTable iii. U 


~~ 


56° 


57° 


58° 


59° 


60° 


61° 


62° 


63° 


' 





829 0376 


838 6706 


848 0481 


857 1673 


866 0254 


874 6197 


882 9476 


891 0065 


60 


1 


2002 


8290 


2022 


3171 


1708 


7607 


383 0841 


1385 


59 


2 


3628 


9873 


3562 


4668 


3161 


9016 


2206 


2705 


58 


3 


5252 


839 1455 


5102 


6164 


4614 


875 0425 


3569 


4024 


57 


4 


6877 


3037 


6641 


7660 


6066 


1832 


4933 


5342 


56 


5 


8500 


4618 


8179 


9155 


7517 


3239 


6295 


6659 


55 ' 


6 


830 0123 


6199 


9717 


858 0649 


8967 


4645 


7656 


7975 


54 1 


7 


1745 


7778 


849 1254 


2143 


867 0417 


6051 


9017 


9291 


53 ! 


8 


3366 


9357 


2790 


3635 


1866 


7455 


884 0377 


892 0606 


52 


9 


4987 


840 0936 


4325 


5127 


3314 


8859 


1736 


1920 


51 


10 


6607 


2513 


5860 


6619 


4762 


876 0263 


3095 


3234 


50 , 


11 


8226 


4090 


7394 


8109 


6209 


1665 


4453 


4546 


49 


12 


9845 


5666 


8927 


9599 


7655 


3067 


5810 


5858 


48 


13 


831 1463 


7241 


850 0459 


859 1088 


9100 


4468 


7166 


7169 


47 


14 


3080 


8816 


1991 


2576 


868 0544 


5868 


8522 


8480 


46 


15 


4696 


841 0390 


3522 


4064 


1988 


7268 


9876 


9789 


45 


16 


6312 


1963 


5053 


5551 


3431 


8666 


885 1230 


893 1098 


44 


17 


7927 


3536 


6582 


7037 


4874 


877 0064 


2584 


2406 


43 


18 


9541 


5108 


8111 


8523 


6315 


1462 


3936 


3714 


42 


19 


832 1155 


6679 


9639 


860 0007 


7756 


2858 


5288 


5Q21 


41 


20 


2768 


8249 


851 1167 


1491 


9196 


4254 


6639 


6326 


40 


21 


4380 


9819 


2693 


2975 


869 0636 


5649 


7989 


7632 


39 


22 


5991 


842 1388 


4219 


' 4457 


2074 


7043 


9339 


8936 


38 


23 


7602 


2956 


5745 


5939 


3512 


8437 


886 0688 


894 0240 


37 


24 


9212 


4524 


7269 


7420 


4949 


9830 


2036 


1542 


36 


25 


833 0822 


6091 


8793 


8901 


6386 


878 1222 


3383 


2844 


35 


26 


2430 


7657 


852 0316 


861 0380 


7821 


2613 


4730 


4146 


34 


27 


4038 


9222 


1839 


1859 


9256 


4004 


6075 


5446 


33 


28 


5646 


843 0787 


3360 


3337 


870 0691 


5394 


7420 


6746 


32 


29 


7252 


2351 


4881 


4815 


2124 


6783 


8765 


8045 


31 


30 


8858 


3914 


6402 


6292 


3557 


8171 


887 0108 


9344 


30 


31 


B34 0463 


5477 


7921 


7768 


4989 


9559 


1451 


895 0641 


29 


32 


2068 


7039 


9440 


9243 


6420 


879 0946 


2793 


1938 


28 


33 


3672 


8600 


853 0958 


862 0717 


7851 


2332 


4134 


3234 


27 


34 


5275 


844 0161 


2475 


2191 


9281 


3717 


5475 


4529 


26 


35 


6877 


1720 


3992 


o864 


871 0710 


5102 


6815 


5824 


25 


36 


8479 


3279 


5508 


5137 


2138 


6486 


8154 


7118 


24 '; 


37 


835 0080 


4838 


7023 


6608 


3566 


7869 


9492 


8411 


23 


1 38 


1680 


6395 


8538 


8079 


4993 


9251 


888 0830 


9703 


22 i 


1 39 


3279 


7952 


854 0051 


9549 


6419 


880 0633 


2166 


896 0994 


21 


;40 


4878 


9508 


1564 


863 1019 


7844 


2014 


3503 


2285 


20 , 


41 


6476 


845 1064 


3077 


2488 


9269 


3394 


4838 


3575 


19 1 


42 


8074 


2618 


4588 


3956 


872 0693 


4774 


6172 


4864 


18 


43 


9670 


4172 


6099 


5423 


2116 


6152 


7506 


6153 


17 


44 


836 1266 


5726 


7609 


6889 


3538 


7530 


8839 


7440 


16 


45 


2862 


7278 


9119 


8355 


4960 


8907 


889 0171 


8727 


15 


46 


4456 


8830 


855 0627 


9820 


6381 


881 0284 


1503 


897 0014 


14 


47 


6050 


846 0381 


2135 


864 1284 


7801 


1660 


2834 


1299 


13 


48 


7643 


1932 


3643 


2748 


9221 


3035 


4164 


2584 


12 


49 


9236 


3481 


5149 


4211 


873 0640 


4409 


5493 


3868 


11 


50 


837 0827 


5030 


6655 


5673 


2058 


5782 


6822 


5151 


10 


51 


2418 


6579 


8160 


7134 


3475 


7155 


8149 


6433 


9 


52 


4009 


8126 


9664 


8595 


4891 


8527 


9476 


7715 


8 


53 


5598 


9673 


856 1168 


865 0055 


6307 


9898 


890 0803 


8996 


7 


54 


7187 


847 1219 


2671 


1514 


7722 


882 1269 


2128 


898 0276 


6 


55 


8775 


2765 


4173 


2973 


9137 


2638 


3453 


1555 


5 


56 


838 0363 


4309 


5674 


4430 


874 0550 


4007 


4777 


2834 


4 


57 


1950 


5853 


7175 


5887 


1963 


5376 


6100 


4112 


3 


58 


3536 


7397 


8675 


7344 


3375 


6743 


7423 


5389 


2 


59 


5121 


8939 


857 0174 


8799 


4786 


8110 


8744 


6665 


1 


60 


6706 


848 0481 


1673 


866 0254 


6197 


9476 


891 0065 


7940 





/ 


33° 


32° 


31° 


30° 


29° 


28° 


27° 


26° 


' 


' ,. 


NAT. COSINE. i| 



Table in. 






NAT. TAN. 


177 




' 


56° 


57^ 


f 580 


5'Jo 


60° 


61° 


620 


63° . 


' 







1 -43256 10 


I •5-^98650 


1-6003345 


1-664-2795 


1-7320508 


1-8040478 


1-8807265 


1-9626105 


60 




1 


34916 


1-5408460 


13709 


53766 


32149 


52860 


20470 


40227 


59 




2 


44231 


18280 


24082 


64748 


43803 


65256 


33690 


54364 


58 




3 


53554 


28108 


34465 


75741 


65468 


77664 


46924 


68518 


57 




4 


62884 


37946 


44858 


86744 


67144 


90086 


60172 


82688 


56 




5 


72223 


47792 


55260 


97758 


78833 


1-8102521 


73436 


96874 


55 




6 


81570 


57647 


65672 


1-6708782 


90533 


14969 


86713 


1-9711077 


54 




7 


90925 


67510 


76094 


19818 


1-7402245 


27430 


1-8900006 


25296 


53 




8 


1-4900288 


77383 


86525 


30864 


13969 


39904 


13313 


39531 


52 




9 


09659 


87264 


96966 


41921 


25705 


52391 


26635 


53782 


51 




10 


19039 


97155 


1-6107417 


52988 


37453 


64892 


39971 


68050 


50 




11 


28426 


1-5507054 


17878 


64067 


49213 


77405 


53322 


82334/49 




12 


37S22 


16963 


28349 


75156 


60984 


89932 


66688 


96635 


48 




13 


47225 


26880 


38829 


86256 


72768 


1-8202473 


80068 


1-9810952 


47 




14 


56637 


36806 


49320 


97367 


84564 


15026 


93464 


25286 


46 




15 


66058 


46741 


59820 


1-6808489 


96371 


27593 


1-9006874 


39636 


45 




16 


75486 


56685 


70330 


19621 


1-7508191 


40173 


20299 


54003 


44 




17 


84923 


66639 


80850 


30765 


20023 


52767 


33738 


68387 


43 




18 


94367 


76601 


91380 


41919 


31866 


65374 


47193 


82787 


42 




19 


1-5003821 


86572 


1-6201920 


53085 


43722 


77994 


60663 


97204 


41 




20 


13282 


96552 


12469 


64261 


55590 


90628 


74147 


1-9911037 


40 




21 


22751 


1-5606542 


23029 


75449 


67470 


1-8303275 


87647 


26087 


39 




22 


32229 


16540 


33599 


86647 


79362 


15936 


1-9101162 


40554 


38 




23 


41716 


26548 


44178 


97856 


91267 


28610 


14691 


55038 


37 




24 


51210 


36564 


54768 


1-6909077 


1-7603183 


41297 


28236 


69539 


36 




25 


60713 


46590 


65368 


20308 


15112 


53999 


41795 


84056 


35 




26 


70224 


56625 


75977 


31550 


27053 


66713 


55370 


98590 


34 




27 


79743 


66669 


86597 


42804 


39007 


79442 


68960 


2-0013142 


33 




28 


89271 


76722 


97227 


54069 


50972 


92184 


82565 


27710 


32 




29 


98807 


86784 


1-6307867 


65344 


62950 


1-8404940 


96186 


42295 


31 




30 


1-5108352 


96856 


18517 


76631 


74940 


17709 


1-9209821 


56897 


30 




31 


17905 


1-5706936 


29177 


87929 


86943 


30492 


23472 


71516 


29 




32 


27466 


17026 


39847 


99238 


98958 


43289 


37138 


86153 


28 




33 


37036 


27126 


50528 


1-7010559 


1-7710985 


56099 


50819 


2-0100806 


27 




34 


46614 


37234 


61218 


21890 


23024 


68923 


64516 


15477 


26 




35 


56201 


47352 


71919 


33233 


35076 


81761 


78228 


30164 


25 




36 


65796 


57479 


82630 


44587 


47141 


94613 


91956 


44869 


24 




37 


75400 


67615 


93351 


55953 


59218 


1-8507479 


1-9305699 


59592 


23 




38 


85012 


77760 


1-6404082 


67329 


71307 


20358 


19457 


74331 


22 




39 


94632 


87915 


14824 


78717 


83409 


33252 


33231 


89088 


21 




40 


1-5204261 


98079 


25576 


90116 


95524 


46159 


47020 


2-0203862 


20 




41 


- 13899 


1-5808253 


36338 


1-71015-27 


17807651 


59080 


60825 


18654 


19 




42 


23545 


18436 


47111 


12949 


19790 


72015 


74645 


33462 


18 




43 


33200 


28628 


57893 


24382 


31943 


84965 


88481 


48289 


17 




44 


42863 


38830 


68687 


35827 


44107 


97928 


1-9402333 


63133 


16 




45 


52535 


49041 


79490 


47283 


56285 


1-8610905 


16200 


77994 


15 




46 


62215 


59261 


90304 


58751 


68475 


23896 


30083 


92873 


14 




47 


71904 


69491 


1-6501128 


70230 


80678 


36902 


43981 


2-0307769 


13 




48 


81602 


79731 


11963 


81720 


92893 


49921 


57896 


22683 


12 




49 


91308 


89979 


22808 


93222 


1-7905121 


62955 


71826 


37615 


11 




50 


1-5301023 


1-5900238 


33663 


1-7204736 


17362 


76003 


85772 


52565 


10 




51 


10740 


10505 


44529 


16261 


29616 


89065 


99733 


67532 


9 




52 


20479 


20783 


55405 


27797 


41883 


1-8702141 


1-9513711 


82517 


8 




53 


30219 


31070 


66292 


39346 


54162 


15231 


27704 


97519 


7 




54 


39969 


41366 


77189 


50905 


66454 


28336 


41713 


2-0412540 


6 




55 


49727 


51672 


88097 


62477 


78759 


41455 


55739 


27578 


5 




56 


59494 


61987 


99016 


74060 


91077 


54588 


69780 


42634 


4 




57 


69270 


72312 


1-6609945 


85654 


1-8003408 


67736 


83837 


57706 


3 




58 


79054 


82647 


20884 


97260 


15751 


80898 


97910 


72800 


2 




59 


88848 


92991 


31834 


1-7308878 


28108 


94074 


1-9612000 


87910 


1 




60 


98650 


1-6003345 


42795 


20508 


40478 


1-8807265 


26105 


2-0503038 







' 


330 


320 


31° 


30O 


29° 


28° 


270 


260 


'' 





NAT. COTAN. 



178 


NAT. SINE. 


[Table m. 




1 


64° 1 65° 


66° 


67° 


68° 


69° 


70° 


71° 


' 







898 79401906 3078 


913 5455 


920 5049 


927 1639 


933 5804 


939 6926 


945 5186 


60 




1 


9215 


4307 


6637 


6185 


2926 


6846 


7921 


6132 


59 




2 


899 0489 


5535 


7819 


7320 


4016 


7888 


8914 


7078 


58 




! 3 


1763 


6762 


9001 


8455 


5104 


8928 


9907 


8023 


57 




4 


3035 


7989 


914 0181 


9569 


6191 


9968 


940 0899 


8968 


56 




5 


4307 


9215 


1361 


921 0722 


7277 


934 1007 


1891 


9911 


55 




6 


5578 


907 0440 


2540 


1854 


8363 


2045 


2881 


946 0854 


54 




7 


6848 


1665 


3716 


2966 


9447 


3082 


3871 


1795 


53 




8 


6117 


2886 


4895 


4116 


928 0531 


4119 


4860 


2736 


52 




9 


9386 


4111 


6072 


5246 


1614 


5154 


5848 


3677 


51 




10 


900 0654 


5333 


7247 


6375 


2696 


6189 


6835 


4616 


50 




11 


1921 


6554 


8422 


7504 


3776 


7223 


7822 


5555 


49 




12 


3188 


7775 


9597 


8632 


4858 


8257 


8608 


6493 


48 




! 13 


4453 


8995 


915 0770 


9756 


5938 


9289 


9793 


7430 


47 




14 


5718 


908 0214 


1943 


922 0884 


7017 


935 0321 


941 0777 


8366 


46 




; 15 


6982 


1432 


3115 


2010 


8096 


1352 


1760 


9301 


45 




16 


8246 


2649 


4266 


3134 


9173 


2382 


2743 


947 0236 


44 




17 


9508 


3866 


5456 


4258 


929 0250 


3412 


3724 


1170 


43 




18 


901 0770 


5082 


6626 


5361 


1326 


4440 


4705 


2103 


42 




19 


2031 


6297 


7795 


6503 


2401 


5468 


5686 


3035 


41 




20 


3292 


7511 


8963 


7624 


3475 


6495 


6665 


3966 


40 




21 


4551 


8725 


916 0130 


8745 


4549 


7521 


7644 


4897 


39 




22 


5810 


9938 


1297 


9865 


5622 


8547 


8621 


5827 


38 




123 


7068 


909 1150 


2462 


923 0984 


6694 


9571 


9598 


6756 


37 




24 


8325 


2361 


3627 


2102 


7765 


936 0595 


942 0575 


7684 


36 




25 


9582 


3572 


4791 


3220 


8835 


1618 


1550 


8612 


35 




26 


902 0838 


4761 


5955 


4336 


9905 


2641 


2525 


9538 


34 




27 


2092 


5990 


7118 


5452 


930 0974 


3662 


3498 


948 0464 


33 




28 


3347 


7199 


8279 


6567 


2042 


4683 


4471 


1389 


32 




29 


4600 


8406 


9440 


7682 


3109 


5703 


5444 


2313 


31 




30 


5853 


9613 


917 0601 


8795 


4176 


6722 


6415 


3237 


30 




31 


7105 


910 0819 


1760 


9908 


5241 


7740 


7386 


4159 


29 




32 


8356 


2024 


2919 


924 1020 


6306 


8758 


8355 


5081 


28 




33 


9606 


3228 


4077 


2131 


7370 


9774 


9324 


6002 


27 




34 


903 0856 


4432 


5234 


3242 


8434 


937 0790 


943 0293 


6922 


26 




35 


2105 


5635 


6391 


4351 


9496 


1806 


1260 


7842 


25 




36 


3353 


6837 


7546 


5460 


931 0558 


2820 


2227 


8760 


24 




37 


4600 


6038 


8701 


6568 


1619 


3833 


3192 


9678 


23 




38 


5847 


9238 


9855 


7676 


2679 


4346 


4157 


949 0595 


22 




39 


7093 


911 0438 


918 1009 


8782 


3739 


5856 


5122 


1511 


21 




40 


8338 


1637 


2161 


9686 


4797 


6869 


6085 


2426 


20 




41 


9582 


2835 


3313 


925 0993 


5855 


7680 


7048 


3341 


19 




42 


904 0825 


4033 


4464 


2097 


6912 


8889 


8010 


4255 


18 




43 


2068 


5229 


5614 


3201 


7969 


9898 


8971 


5168 


17 




44 


3310 


6425 


6763 


4303 


9024 


938 0906 


9931 


6080 


16 




45 


4551 


7620 


7912 


5405 


932 0079 


1913 


944 0890 


6991 


15 




46 


5792 


6815 


906C 


6506 


1133 


2920 


1649 


7902 


14 




47 


7032 


912 0008 


919 0207 


7606 


2186 


3925 


2807 


8812 


13 




48 


8271 


1201 


1353 


8706 


3238 


4930 


3764 


9721 


12 




49 


9509 


2393 


2499 


9805 


4290 


5934 


4720 


950 0629 


11 




50 


905 0746 


3584 


3644 


926 0902 


5340 


6938 


5675 


1536 


10 




51 


1983 


4775 


4788 


2000 


6390 


7940 


6630 


2443 


9 




52 


3219 


5965 


5931 


3096 


7439 


8942 


7584 


3348 


8 




53 


4454 


7154 


7073 


4192 


8488 


9943 


8537 


4253 


7 




54 


5688 


8342 


8215 


5266 


9535 


939 0943 


9489 


5157 


6 




55 


6922 


9529 


9356 


6380 


933 0582 


1942 


945 0441 


6061 


5 




56 


8154 


913 0716 


920 0496 


7474 


1628 


2940 


1391 


6963 


4 




57 


9386 


1902 


1635 


8566 


2673 


3938 


2341 


7865 


3 




58 


906 0618 


3087 


2774 


9658 


3718 


4935 


3290 


8766 


2 




59 


1848 


4271 


3912 


927 0748 


4761 


5931 


4238 


9666 


I 




60 


3078 


5455 


5049 


1839 


5604 


6926 


5186 


951 0565 







/ 


25° 


24° 


23° 


22° 


21° 


20° 


19° 


18° 






NAT. COSINE. 


1 





Table iii.] nat. tan. 179 \ 


■' 


64° 


65° 


&&^ 


67° 


68° 


69° 


70° 


71° 


/ 





20503038 


2-1445069 


2-2460368 


2-3558524 


2-4750869 


2-6050891 


2-7474774 


2-9042109 


60 


1 


18185 


61366 


77962 


77590 


71612 


73558 


99661 


69576 


59 


2 


33349 


77683 


95580 


96683 


92386 


96259 


2-7524588 


97089 


58 


3 


48531 


94021 


2-2513221 


2-3615801 


2-4813190 


2-6118995 


49554 


2-9124649 


57 


4 


63732 


2-1510378 


30885 


34946 


34023 


41766 


74561 


52256 


56 


5 


78950 


26757 


48572 


54118 


54887 


64571 


99608 


79909 


55 


6 


94187 


43156 


66283 


73316 


75781 


87411 


2-7624695 


2-9207610 


54 


7 


2'0609442 


59575 


84016 


92540 


96706 


2-6210286 


49822 


35358 


53 


8 


24716 


76015 


2-2601773 


2-3711791 


2-4917660 


33196 


74990 


63152 


52 


9 


40008 


92476 


19554 


31068 


38645 


56141 


2-7700199 


90995 


51 


10 


55318 


2-1608958 


37357 


50372 


59661 


79121 


25448 


2-9318885 


50 


11 


70646 


25460 


55184 


69703 


80707 


2 6302136 


50738 


46822 


49 


12 


85994 


41983 


73035 


89060 


2-5001784 


25186 


76069 


74807 


48 


13 


20701359 


58527 


90909 


2-3808444 


22891 


48271 


2-7801440 


2-9402840 


47 


14 


16743 


75091 


2-2708807 


27855 


44029 


71392 


26853 


30921 


46 


15 


32146 


91677 


26729 


47293 


65198 


94549 


52307 


59050 


45 


16 


47567 


2-1708283 


44674 


66758 


86398 


2-6417741 


77802 


87227 


44 


17 


63007 


24911 


62643 


86250 


2 5107629 


40969 


2-7903339 


2-9515453 


43 


18 


78465 


41559 


80636 


2-3905769 


28890 


64232 


28917 


43727 


42 


19 


93942 


58229 


98653 


25316 


50163 


87531 


54537 


72050 


41 


20 


20809438 


74920 


2-2816693 


44889 


71507 


2-6510867 


80198 


2-9600422 


40 


21 


24953 


91631 


34758 


64490 


92863 


34238 


2-8005901 


28842 


39 


22 


40487 


2-1808364 


52846 


84118 


2-5214249 


57645 


31646 


57312 


38 


23 


56039 


25119 


70959 


2-4003774 


35667 


81089 


57433 


'85831 


37 


24 


71610 


41894 


89096 


23457 


57117 


2-6604569 


83263 


2-9714399 


36 


25 


87200 


58691 


2-2907257 


43168 


78598 


28085 


2-8109164 


43016 


35 


26 


20902809 


75510 


25442 


62906 


2-5300111 


51638 


35048 


71683 


34 


27 


18437 


92349 


43651 


82672 


21655 


75227 


61004 


2-9800400 


33 


28 


34085 


21909210 


61885 


2-4102465 


43231 


98853 


87003 


29167 


32 


29 


49751 


26093 


80143 


22286 


64839 


2-6722516 


2-8213045 


57983 


31 


30 


65436 


42997 


98425 


42136 


86479 


46215 


39129 


86850 


30 


31 


81140 


59923 


2-3016732 


62013 


2-5408151 


69951 


65256 


2-9915766 


29 


32 


96864 


76871 


35064 


81918 


29855 


93725 


91426 


44734 


28 


33 


21012607 


93840 


53420 


2-4201851 


51591 


2-6817535 


2-8317639 


73751 


27 


34 


28369 


2.2010831 


71801 


21812 


73359 


41383 


43896 


3-0002820 


26 


35 


44150 


27843 


90206 


41801 


95160 


65267 


70196 


31939 


25 


36 


59951 


44878 


2-3108637 


61819 


2-5516992 


89190 


96539 


61109 


24 


37 


75771 


61934 


27092 


81864 


38858 


2-6913149 


2-8422926 


90330 


23 


38 


91611 


79012 


45571 


2-4301938 


60756 


37147 


49356 


3-0119603 


22 


39 


2-1107470 


96112 


64076 


22041 


82686 


61181 


75831 


48926 


21 


40 


23348 


2-2113234 


82606 


42172 


2-5604649 


85254 


2-8502349 


78301 


20 


41 


39246 


30379 


2-3201160 


62331 


26645 


2-7009364 


28911 


3-0207728 


19 


42 


55164 


47545 


19740 


82519 


48674 


33513 


55517 


37207 


18 


43 


71101 


64733 


38345 


2-4402736 


70735 


57699 


82168 


66737 


17 


44 


87057 


81944 


56975 


22982 


92830 


81923 


2-8608863 


96320 


16 


45 


2-1203034 


99177 


75630 


43256 


2-5714957 


2-7106186 


35602 


3-0325954 


15 


46 


19030 


2-2216432 


94311 


63559 


37118 


30487 


62386 


55641 


14 


47 


35046 


33709 


2-3313017 


83891 


59312 


54826 


89215 


85381 


13 


48 


51082 


51009 


31748 


2-4504252 


81539 


79204 


2-8716088 


3-0415173 


12 


49 


67137 


68331 


50505 


24642 


2-5803800 


2-7203620 


43007 


45018 


11 


50 


83213 


85676 


69287 


45061 


26094 


28076 


69970 


74915 


10 


51 


99308 


i-2303043 


88095 


65510 


48421 


52569 


96979 


3-0504866 


9 


52 


21315423 


20433 


2-3406928 


85987 


70782 


77102 


2-8824033 


34870 


8 


53 


31559 


37845 


25787 


^-4606494 


93177 


2-7301674 


51132 


64928 


7 


54 


47714 


55280 


44672 


27030 


2-5915606 


26284 


78277 


95038 


6 


55 


63890 


72738 


63582 


47596 


38068 


50934 


2-8905467 


30625203 


5 


56 


80085 


90218 


82519 


68191 


60564 


75623 


32704 


55421 


4 


57 


96301 


2-2407721 


2-3501481 


88816 


83095 


2-7400352 


59986 


85694 


3 


58 


21412537 


25247 


20469 


2-4709470 


2-6005659 


25120 


87314 


3-0716020 


2 


59 


28793 


42796 


39483 


30155 


28258 


49927 


2-9014688 


46400 


1 


60 


45069 


60368 


58524 


50869 


50891 


74774 


42109 


76835 





/ 


25° 


24° 


23° 


22° 


21° 


20° 


19° , 18° 1 


' 


NAT. COTAN. 1 



180 


NAT. SINE. 


{Table m.jj 


/ 


72° 


73° 


74° 


75° 1 


76° 1 


77° 1 


78° 1 


79° 1 


/ 





}51 0565 


356 3048 \ 


)61 2617 \ 


365 9258 970 29571974 37011978 1476J981 6272| 


60 


1 


1464 


3898 


3418 


366 0011 


3661 


4355 


2080 


6826 


59 


2 


2361 


4747 


4219 


0762 


4363 


5008 


2684 


7380 


58 


3 


3258 


5595 


5019 


1513 


5065 


5660 


32S7 


7933 


57 


4 


4154 


6443 


5818 


2263 


5766 


6311 


3889 


8485 


56 


5 


5050 


7290 


6616 


3012 


6466 


6962 


4490 


9037 


55 


6 


5944 


8136 


7413 


3761 


7165 


7612 


5090 


9587 


54 


7 


6838 


8981 


8210 


4508 


7863 


8261 


5089 982 01371 


53 


8 


7731 


9825 


9005 


5255 


8561 


8909 


6288 


0686 


52 


9 


8623 


357 0669 


9800 


6001 


9258 


9556 


6886 


1234 


51 


10 


9514 


1512 


362 0594 


67461 


9953 975 0203] 


7483 


1781 


50 


11 


352 0404 


2354 


1387 


7490 


371 0649 


0849 


8079 


2327 


49 


12 


1294 


3195 


2180 


8234 


1343 


1494 


8674 


2873 


48 


13 


2183 


4035 


2972 


8977 


2036 


2133 


9268 


3417 


47 


14 


3071 


4875 


3762 


9718 


2729 


2781 


9862 3961 


46 


15 


3958 


5714 


4552 


967 0459 


3421 


3423 


379 0455 4504 


45 


16 


4844 


6552 


5342 


1200 


4112 


4065 


1047 


5046 


44 


17 


5730 


7389 


6130 


1939 


4802 


4706 


1638 


5587 


43 


18 


6615 


8225 


6917 


2678 


5491 


5345 


2228 


6128 


42 


19 


7499 


9060 


7704 


3415 


6180 5985 


2818 


6668 


41 


20 


8382 


9895 


8490 


4152 


6867 6623 


3406 


7206 


40 


21 


9264 


958 0729 


9275 


4888 


7554 7260 


3994 


7744 


39 


22 


953 0146 


1562 


963 0060 


5624 


8240 


7897 


4581 


8282 


38 


23 


1027 


2394 


0843 


6358 


8926 


8533 


5167 


8818 


37 


24 


1907 


3226 


1626 


7092 


9610 


9168 


5752 


9353 


36 


25 


2786 


4056 


2408 


7825 


972 0294 


9802 


6337 


9888 


35 


26 


3664 


4886 


3189 


8557 


0976 


976 0435 


6921 


983 0422 


34 


27 


4542 


5715 


3969 


9288 


1658 


1068 


7504 


0955 


33 


28 


5418 


6543 


4748 


968 0018 


2339 


1699 


8086 


1487 


32 


29 


6294 


7371 


5527 


0748 


3020 


2330 


8668 


2019 


31 


30 


7170 


8197 


6305 


1476 


3699 


2960 


9247 


2549 


30 


31 


8044 


9023 


7081 


2204 


4378 


3589 


9827 


3079 


29 


32 


8917 


9848 


7858 


2931 


5056 


4218 


980 0405 


3608 


28 


33 


9790 


959 0672 


8633 


3658 


5733 


4845 


0983 


4136 


27 


34 


954 0662 


1496 


9407 


4383 


6409 


5472 


1560 


4663 


26 


35 


1533 


2318 


954 0181 


5108 


7084 


6098 


2136 


5189 


25 


36 


2403 


3140 


0954 


5832 


7759 


6723 


2712 


5715 


24 


37 


3273 


3961 


1726 


6555 


8432 


7347 


3286 


6239 


23 


38 


4141 


4781 


2497 


7277 


9105 


7970 


3860 


6763 


22 


39 


5009 


5600 


3268 


7998 


9777 


8593 


4433 


7286 


21 


40 


5876 


6418 


4037 


8719 


973 0449 


9215 


5005 


760E 


20 


41 


6743 


7236 


4806 


9438 


1119 


9836 


5576 


833C 


19 


42 


7608 8053 


5574 


969 0157 


1789 


977 0456 


6147 


885C 


18 


43 


8473 8869 


6341 


0875 


2458 


1075 


67ie 


937C 


17 


44 


9336 


9684 


710S 


1593 


3125 


1693 


7285 


9889 


) 16 


45 


955 0199 


960 0499 


7872 


2309 


3793 


2311 


785S 


984 040' 


^ 15 


46 


1062 


1315 


8635 


302= 


4459 


292E 


842C 


092^ 


I 14 


47 


192S 


212= 


940-^ 


I 374C 


1 5124 


3544 


898f 


) 144] 


13 


48 


278^ 


293' 


' 965 016J 


) 445: 


}' 5789 


4159 


95K 


\ 195( 


5 12 


49 


364: 


\ 374f 


\ 092' 


5167 645: 


\ 477: 


i 981 OIK 


) 247 


L 11 


50 


450- 


I 4555 


5 163 


3 5879 71 H 


) 538' 


r 068( 


) 298 


5 10 


51 


536 


L 5365 


^ 244 


3 659 


I 7775 


5 599J 


) 124: 


5 349 


3 9 


52 


621 


3 617- 


r 320 


3 730 


1 843' 


3 661 


I 180. 


5 401 


3 8 


53 


707^ 


i 698^ 


1 396 


3 801 


1 910( 


3 722' 


I 236( 


5 452 


1 7 


54 


793 


3 779 


2 472 


6 872 


976 


3 783 


I 292 


r 503 


2 6 


55 


878 


5 859 


3 548 


4 942 


8 974 041 


3 844 


I 348 


3 554 


2 5 


56 


963 


9 940 


3 624 


970 013 


6 107 


7 905 


3 404 


5 605 


4 


57 


956 049 


2 961 020 


B 699 


6 084 


2 173 


4 965 


8 460 


3 655 


8 3 


58 


134 


5 101 


2 775 


1 154 


8 23901978 026 


5 516 


706 


6 2 


59 


219 


7 181 


5 650 


5 225 


3 304 


6 087 


1 571 


6 757 


2 1 


60 


304 


8 261 


7 925 


8 295 


7 370 


1 147 


6 6272 807 


8 


' 


17° 


16=^ 


15° 


14° 


13° 


12° 


11° 10° 1 ' 11 


__ 


NAT. COSINE. 


Ji 



Table iii.] 



NAT. TAN. 



181 



72° 

3-077683 

3-0807325 

37869 

68468 

99122 

3-0929831 

60596 

91416 

3-1022291 

53223 

84210 

3-1115254 
46353 
77509 

3-1208722 
39991 
71317 

3-1302701 
34141 
65639 
97194 

3-1428807 
60478 
92207 

3-1523994 
55840 
87744 

3-1619706 
51728 
83808 

3-1715948 
48147 
80406 

3-1812724 
45102 
77540 

3-1910039 
42598 
75217 

3-2007897 

40638 
73440 

3-2106304 
39228 
72215 

3-2205263 
38373 
71546 

3-2304780 
38078 



71438 

3-2404860 
38346 
71895 

3-2505508 
39184 
72924 

3-2606728 
40596 
74529 

3-2708526 
170 



730 

3-2708526 
42588 
76715 

3-2810907 
45164 
79487 

3-2913876 
48830 
82851 

3-3017438 

52091 
86811 

3-3121598 
56452 
91373 

3-3226362 
61419 
96543 

3-3331736 
66997 

3-3402326 
37724 
73191 

3-3508728 
44333 
80008 

3-3615753 
51568 
87453 

3-3723408 
59434 
95531 

3-3831699 
67938 

3-3904249 
40631 
77085 

3-4013612 
50210 



3-4123626 
60443 
97333 

3-4234297 
71334 

3-4308446 
45631 
82891 

3-4420226 
57635 
95120 

3-4532679 
70315 

3-4608026 
45813 
83676 

3-4721616 
59632 
97726 

3-4835896 
74144 
16° 



740 

3-4874144 

3-4912470 
50874 
89356 

3-5027916 
66555 

3-5105273 
44070 
82946 

3-5221902 

60938 

3-5300054 
39251 
78528 

3-5417886 
57325 
96846 

3-5526449 
76133 

3-5615900 

55749 
95681 

3-5735696 
75794 

3-5815975 
56241 
96590 

3-5937024 
77543 

3-6018146 



99609 
3-6140469 

81415 
3-6222447 

63566 
3-6304771 

46064 

87444 
3-6428911 

70467 
3-6512111 

53844 

95665 
3-6637575 

79575 
3-6721665 

63845 
3-6806115 

48475 

90927 
3-6933469 

76104 
3-7018830 

61648 
3-7104558 

47561 

90658 
3-7233847 

77131 
3-7320508 

15° 



75 
3-7320508 

63980 
3-7407546 

51207 

94963 
3-7538815 

82763 
3-7626807 

70947 
3-7715185 

59519 
3-7803951 

48481 

93109 
3-7937835 

82661 
3-8027585 

72609 
3-8117733 

62957 

3-8208281 
53707 
99233 

3-8344861 
90591 

3-8436424 
82358 

3-8528396 
74537 

3-8620782 

67131 

3-8713584 
60142 

3-8806805 
53574 

3-8900448 
47429 
94516 

3-9041710 
89011 

3-9136420 

83937 
3-9231563 

79297 
3-9327141 

75094 
3-9423157 

71331 
3-9519615 

68011 



3-9616518 

65137 
3-9713868 

62712 
3-9811669 

60739 
3-9909924 

59223 
4-0008636 

58165 
4-0107809 
14° 



76° 
■0107809 

57570 
-0207446 

57440 
•0307550 

57779 
-0408125 

58590 
-0509174 

59877 

•0610700 

61643 
-0712707 

63892 
-0815199 

66627 
-0916178 

69852 
•1021649 

73569 
-1125614 

77784 4 
-1230079 

82499 
-1335046 

87719 
-1440519 

93446 
-1546501 

99685 



4-1652998 
4-1706440 

60011 
4-1813713 

67546 
4-1921510 

75606 
4-2029835 

84196 
4-2138690 

93318 
4-2248080 
4-2302977 

58009 
4-2413177 

68432 
4-2523923 

79501 
4-2635218 

91072 



■2747066 
2803199 

59472 
•2915885 

72440 
■3029136 

85974 
3142955 
3200079 

57347 
3314759 
130 



770 

-3314759 

72316 
-3430018 

87866 
•3545861 
-3604003 

62293 
-3720731 

79317 
-3838054 

96940 
-3955977 
-4015164 

74504 
•4133996 

93641 
-4253439 
-4313392 

73500 
■4433762 

94181 
■4554756 
■4615489 

76379 
■4737428 

98636 
4860004 
4921532 

83221 
'5045072 
5107085 

69261 
5231601 

94105 
5356773 
5419608 



4-5545776 

4-5609111 

72615 

4-5736287 
4-5800129 

64141 
4-5928325 

92680 
4-6057207 
4-6121908 

86783 
4-6251832 
4-6317056 

82457 
4-6448034 
4-6513788 

79721 
4-6645832 
4-6712124 

78595 
4-6845248 
4-6912083 

79100 
4-7046301 
12° I 



78° 

■7046301 

■7113686 

81256 

■7249012 

■7316954 

85083 

■7453401 

7521907 

90603 

4-7659490 

4-7728568 

97837 

4-7867300 

4-7936957 

4-8006808 

76854 

4-8147096 

4-8217536 

88174 

4-8359010 



8430045 

8501282 

72719 

8644359 

8716201 

88248 

8860499 

8932956 

9005620 

78491 

4-9151570 

4-9224859 

98358 

4-9372068 

4-9445990 

4-9520125 

94474 

4-9669037 

4-9743817 

4-9818813 

94027 

-9969459 

-0045111 

-0120984 

97078 

•0273395 

■0349935 

•0426700 

•0503690 

80907 

•0658352 
■0736025 
0813928 

92061 
0970426 
1049024 
1127855 
1206921 

86224 
1365763 
1445540 

no 



NAT. COT AN. 



182 NAT. SINE. [Table iii. || 


' 


80° 


81<^ 


820 


83° 


84° 


85° 


86° 


87° 


/ 





9848 078 


9876 883 


5902 681 


9925 462 


3945 219 


9961 947 


9975 641 


9986 295 


60 




582 


9877 338 


3903 085 


816 


523 


9962 200 


843 


447 


59 


2 


9849 086 


792 


489 


9926 169 


825 


452 


9976 045 


598 


58 


3 


589 


9878 245 


891 


521 


9946 127 


704 


245 


748 


57 


4 


9850 091 


697 


3904 293 


873 


428 


954 


445 


898 


56 


5 


593 


9879 148 


694 


9927 224 


729 


9963 204 


645 


9987 046 


55 


6 


9851 093 


599 


3905 095 


573 


9947 028 


453 


843 


194 


54 


7 


593 


9880 048 


494 


922 


327 


701 


9977 040 


340 


53 


8 


9852 092 


497 


893 


9928 271 


625 


948 


237 


486 


52 


9 


590 


945 


3906 290 


618 


921 


9964 195 


433 


631 


51 


10 


9853 037 


9881 392 


687 


965 


9948 217 


440 


627 


775 


50 


11 


583 


838 


3907 083 


9929 310 


513 


685 


821 


919 


49 


12 


9854 079 


9882 284 


478 


655 


807 


929 


9978 015 


9988 061 


48 


13 


574 


728 


873 


999 


9949 101 


9965 172 


207 


203 


47 


14 


9855 058 


9883 172 


3908 266 


9930 342 


393 


414 


399 


344 


46 


15 


561 


615 


659 


685 


685 


655 


589 


484 


45 


16 


9856 053 


9884 057 


3909 051 


9931 026 


976 


895 


779 


623 


44 


17 


544 


498 


442 


367 


9950 266 


9966 135 


968 


761 


43 


18 


9857 035 


939 


832 


706 


556 


374 


9979 156 


899 


42 


19 


524 


9885 378 


3910 221 


9932 045 


844 


612 


343 


9989 035 


41 " 


20 


9858 013 


817 


610 


384 


9951 132 


849 


530 


171 


40 


21 


501 


9886 255 


997 


721 


419 


9967 085 


716 


306 


39 


22 


988 


692 


3911 384 


9933 057 


705 


321 


900 


440 


38 


23 


9859 475 


9837 128 


770 


393 


990 


555 


9980 084 


573 


37 


24 


960 


564 


3912 155 


728 


9952 274 


739 


267 


706 


36 


25 


9860 445 


998 


540 


9934 062 


557 


9963 022 


450 


837 


35 


26 


929 


9888 432 


923 


395 


840 


254 


631 


968 


34 


27 


9861 412 


865 


3913 306 


727 


9953 122 


485 


811 


9990 098 


33 


28 


894 


9889 297 


683 


9935 053 


403 


715 


991 


227 


32 


29 


9862 375 


728 


3914 069 


389 


683 


945 


9981 170 


355 


31 


30 


856 


9890 159 


449 


719 


962 


9969 173 


343 


482 


30 


31 


9863 336 


588 


828 9936 047| 


9954 240 


401 


525 


609 


29 


32 


815 


9891 017 


9915 206 


375 


513 


628 


701 


734 


28 


33 


9864 293 


445 


584 


703 


795 


854 


877 


859 


27 


34 


770 


872 


961 


9937 029 


9955 070 


9970 080 


9982 052 


983 


26 


35 


9865 246 


9892 298 


9916 337 


355 


345 


304 


225 


9991 106 


25 


36 


722 


723 


712 


679 


620 


528 


398 


228 


24 


37 


9866 196 


9893 148 


9917 086 


9938 003 


893 


750 


570 


350 


23 


38 


670 


572 


459 


326 


9956 165 


972 


742 


470 


22 


39 


9367 143 


994 


832 


648 


437 


9971 193 


912 


590 


21 


40 


615 


9894 416 


9918 204 


969 


708 


413 


9983 082 


709 


20 


41 


9868 087 


838 


574 


9939 290 


978 


633 


250 


827 


19 


42 


557 


9895 258 


944 


610 


9957 247 


851 


418 


944 


18 


43 


9869 027 


677 


9919 314 


928 


515 


9972 069 


585 


9992 060 


17 


44 


496 


9896 096 


682 


9940 246 


783 


286 


751 


176 


16 


45 


964 


514 


9920 049 


563 


9958 049 


502 


917 


290 


15 


46 


9870 431 


931 


416 


880 


315 


717 


9984 081 


404 


14 


47 


897 


9897 347 


782 


9941 195 


580 


931 


245 


517 


13 


48 


9871 363 


762 


9921 147 


510 


844 


9973 145 


408 


629 


12 


49 


827 


9893 177 


511 


823 


9959 107 


357 


570 


740 


11 


50 


9872 291 


590 


874 


9942 136 


370 


569 


731 


851 


10 


51 


754 


9899 003 


9922 237 


448 


631 


780 


891 


960 


9 


52 


9873 216 


415 


59c 


760 


892 


990 


9985 050 


9993 069 


8 


53 


678 


826 


95c 


9943 070 


9960 152 


9974 199 


209 


177 


7 


54 


9874 138 


9900 237 


9923 315 


379 


411 


406 


367 


284 


6 


55 


598 


646 


67C 


683 


669 


615 


52^ 


39C 


5 


56 


9875 057 


9901 055 


9924 037 


996 


926 


822 


63C 


495 


4 


57 


514 


462 


39^ 


9944 303 


9961 183 


9975 02s 


83£ 


60C 


3 


58 


972 


869 


751 


609 


43e 


233 


98C 


70-^ 


2 


59 


9876 42E 


9902 2759925 10' 


914 


693 


437 


9986 143 


806 


1 


60 


833 


681 462 9945 219 


947 


641 


29£ 


906 





' 


9^ 


go 70 6° 


50 


40 


3° 


2° 


' 


NAT. COSINE. J| 



Table III.] NAT. TAN. 




183 j 


/ 


79° 


80° 


81° 


82° 


83° 


84° 


85° 1 


/ 





5-1445540 


5-6712818 


6-3137515 


7-1153697 


8-1443464 


9-5143645 11-430052 


60 


1 


525557 


809446 


256601 


304190 


639786 


410613 


468474 


59 


' 2 


605813 


906394 


376126 


455308 


837041 


679068 


507154 


58 


1 3 


686311 


5-7003663 


496092 


607056 


8-2035239 


949022 


546093 


57 


4 


767051 


101256 


616502 


759437 


234384 


9-6220486 


585294 


56 


5 


848035 


199173 


737359 


912456 


434485 


493475 


624761 


55 


6 


929264 


297416 


858665 


7-2066116 


635547 


768000 


664495 


54 


7 


5-2010738 


395988 


980422 


220422 


837579 


9-7044075 


704500 


53 


8 


092459 


494889 


6-4102633 


375378 


8-3040586 


321713 


744779 


52 


9 


174428 


594122 


225301 


530987 


244577 


600927 


785333 


51 


10 


256647 


693688 


348426 


687255 


449558 


881732 


826167 


50 


11 


339116 


793588 


472017 


844184 


655536 


9-8164140 


867282 


49 


12 


421836 


893825 


596070 


7-3001780 


862519 


448166 


908682 


48 


13 


504809 


994400 


720591 


160047 


8-4070515 


733823 


950370 


47 


14 


588035 


5-8095315 


845581 


318989 


279531 


9-9021125 


992349 


46 


15 


671517 


196572 


971043 


478610 


489573 


310088 


12-034622 


45 


16 


755255 


298172 


6-5096981 


638916 


700651 


600724 


077192 


44 


17 


839251 


400117 


223396 


799909 


912772 


893050 


120062 


43 


18 


923505 


502410 


350293 


961595 


8-5125943 


10-018708 


163236 


42 


19 


5-3008018 


605051 


477672 


7-4123978 


340172 


048283 


206716 


41 


20 


092793 


708042 


605538 


287064 


555468 


078031 


250505 


40 


21 


177830 


811386 


733892 


450855 


771838 


107954 


294609 


39 


22 


263131 


915084 


862739 


615357 


989290 


138054 


339028 


38 


23 


348696 


5-9019138 


992080 


780576 


8-6207833 


168332 


383768 


37 


24 


434527 


123550 


6-6121919 


946514 


427475 


198789 


428831 


36 


25 


520626 


228322 


252258 


7-5113178 


648223 


229428 


474221 


35 


26 


606993 


333455 


383100 


280571 


870088 


260249 


519942 


34 


27 


693630 


438952 


514449 


448699 


8-7093077 


291255 


565997 


33 


28 


780538 


544815 


646307 


617567 


317198 


322447 


612390 


32 


29 


867718 


651045 


778677 


787179 


542461 


353827 


659125 


31 


30 


955172 


757644 


911562 


957541 


768874 


385397 


706205 


30 


31 


5-4042901 


864614 


6-7044966 


7-6128657 


996446 


417158 


753634 


29 


32 


130906 


971957 


178891 


300533 


8-82-25186 


449112 


801417 


28 


33 


219188 


6-0079676 


313341 


473174 


455103 


481261 


849557 


27 


34 


307750 


187772 


448318 


646584 


686206 


513607 


898058 


26 


35 


396592 


296247 


583826 


820769 


918505 


546151 


946924 


25 


36 


485715 


405103 


719867 


995735 


8-9152009 


578895 


996160 


24 


37 


575121 


514343 


856446 


7-7171486 


386726 


611841 


13-045769 


23 


38 


664812 


623967 


993565 


348028 


622668 


644992 


095757 


22 


39 


754788 


733979 


6-8131227 


525366 


859843 


678348 


146127 


21 


40 


845052 


844381 


269437 


703506 


9-0098261 


711913 


196883 


20 


41 


935604 


955174 


408196 


882453 


337933 


745687 


248031 


19 


42 


5-5026446 


6-1066360 


547508 


7-8062212 


578867 


779673 


299574 


18 


43 


117579 


177943 


687378 


242790 


821074 


813872 


351518 


17 


44 


209005 


289923 


827807 


424191 


9-1064564 


848288 


403867 


16 


45 


300724 


402303 


968799 


606423 


309348 


882921 


456625 


15 


46 


392740 


515085 


6-9110359 


789489 


555436 


917775 


599799 


14 


47 


485052 


628272 


252489 


973396 


802838 


952850 


563391 


13 


48 


577663 


741865 


395192 


7-9158151 


9-2051564 


988150 


617409 


12 


49 


670574 


855867 


538473 


343758 


301627 


11-023676 


671856 


n 


50 


763786 


970279 


682335 


530224 


553035 


059431 


726738 


10 


51 


857302 


6-2085106 


826781 


717555 


805802 


095416 


782060 


9 


52 


951121 


200347 


971806 


905756 


9-3059936 


131635 


837827 


8 


53 


5-6045247 


316007 


7-0117441 


8-0094835 


315450 


168089 


894045 


7 


54 


139680 


432086 


263662 


284796 


572355 


204780 


950719 


6 


55 


234421 


548588 


410482 


475647 


830663 


241712 


14-007856 


5 


56 


329474 


665515 


557905 


667394 


9-4090384 


278885 


065459 


4 


57 


424838 


782868 


705934 


860042 


351531 


316304 


123536 


3 


58 


520516 


900651 


854573 


8-1053599 


614116 


353970 


182092 


2 


59 


616509 


6-3018866 


7-1003826 


248071 


878149 


391885 


241134 


I 


60 


712818 


137515 


153697 


443464 


9-5143645 


430052 


300666 


(V 


' 


10° 


9° 


8° 


7° 


6° 


5° 


4° 'Ml 


i^s—a 


NAT. COTAN. 




11 



184 NAT. SINE. NAT. TAN. [Table III. II 


' 


88° 


89° 


' 




' 


86° 


87° 


88° 


89° 


' i 





9993 908 


9998 477 


60 







14-300666 


19-081137 


28-636253 


57-289962 


60 


1 


9994 009 


527 


59 




1 


360696 


187930 


877089 


58-261174 


59 


2 


110 


577 


58 




2 


421230 


295922 


29-122005 


59-265872 


58 


3 


209 


625 


57 




3 


482273 


405133 


371106 


60-305820 


57, 


4 


308 


673 


56 




4 


543833 


515584 


624499 


61-382905 


56 


5 


405 


720 


55 




5 


605916 


627296 


832299 


62-499154 


55 


6 


502 


766 


54 




6 


668529 


740291 


30-144619 


63-656741 


54 


7 


598 


812 


53 




7 


731679 


854591 


411580 


64-858008 


53 


8 


693 


856 


52 




8 


795372 


970219 


683307 


66-105473 


52 


9 


788 


900 


51 




9 


859616 


20-037199 


959928 


67-401354 


51 


10 


881 


942 


50 




10 


924417 


205553 


31-241577 


68-750087 


50 


11 


974 


984 


49 




11 


989784 


325308 


528392 


70-153346 


49 


12 


9995 066 


9999 025 


48 




12 


15-055723 


446486 


820516 


71-615070 


48 


13 


157 


065 


47 




13 


122242 


569115 


32-113099 


73-133991 


47 


14 


247 


105 


46 




14 


189349 


693220 


421295 


74-729165 


46 


15 


336 


143 


45 




15 


257052 


818328 


730264 


76-390009 


45 


16 


424 


181 


44 




16 


325358 


945966 


33-045173 


78-126342 


44 


17 


512 


218 


43 




17 


394276 


21-074664 


366194 


79-943430 


43 


18 


599 


254 


42 




18 


463814 


204949 


693509 


81-847041 


42 


19 


684 


289 


41 




19 


533931 


336851 


34-027303 


83-343507 


41 


20 


770 


323 


40 




20 


604784 


470401 


367771 


85-939791 


40 


21 


854 


357 


39 




21 


676233 


605630 


715115 


88-143572 


39 


22 


937 


389 


38 




22 


748337 


74-2569 


35-069546 


90-463336 


38 


23 


9996 020 


421 


37. 




23 


821105 


831251 


431282 


92-908487 


37 


24 


101 


452 


36 




24 


894545 


22-021710 


800553 


95-489475 


36 


25 


182 


482 


35 




25 


968667 


163980 


36-177596 


98-217943 


35 


26 


262 


511 


34 




26 


16-043482 


303097 


562659 


101-10690 


34 


27 


341 


539 


33 




27 


118998 


454096 


956001 


104-17094 


33 


28 


419 


567 


32 




23 


195225 


602015 


37-357892 


107-42648 


32 


29 


497 


593 


31 




29 


272174 


751392 


763613 


110-89205 


31 


30 


573 


619 


30 




30 


349855 


903766 


38-138459 


114-58365 


30 


31 


649 


644 


29 




31 


428279 


23-057677 


617738 


118-54018 


29 


32 


724 


668 


28 




32 


507456 


213666 


39-056771 


122-77396 


28 


33 


798 


602 


27' 




33 


587396 


371777 


505895 


127-32134 


27 


34 


871 


714 


26 




34 


668112 


532052 


965460 


132-21851 


26 


35 


943 


736 


25 




35 


749614 


694537 


40-435837 


137-50745 


25 


36 


9997 015 


756 


24 




36 


831915 


359277 


917412 


143-23712 


24 


37 


086 


776 


23 




37 


915025 


24-026320 


41-410588 


149-46502 


23 


38 


156 


795 


22 




38 


998957 


195714 


915790 


156-25908 


22 


39 


224 


813 


21 




39 


17-083724 


367509 


42-433464 


163-70019 


21 


40 


292 


831 


20 




40 


169337 


• 541758 


964077 


171-38540 


20 


41 


360 


847 


19 




41 


255809 


71851.2 


43-508122 


130-93220 


19 


42 


426 


863 


18 




42 


343155 


897826 


44-066113 


190-98419 


18 


43 


492 


878 


17 




43 


431385 


25-079757 


633596 


•202-21875 


17 


44 


556 


892 


16 




44 


520516 


264361 


45-2-26141 


214-85762 


16 


45 


620 


905 


15 




45 


610559 


451700 


829351 


229-18166 


15 


46 


683 


917 


14 




46 


701529 


641832 


46-448362 


245-55198 


14 


47 


745 


928 


13 




47 


793442 


834823 


47-085343 


264-44030 


13 


48 


807 


939 


12 




48 


886310 


26-030736 


739501 


236-47773 


12 


49 


867 


949 


11 




49 


930150 


229638 


48-412034 


312-52137 


11 


50 


927 


958 


10 




50 


13-074977 


431600 


49-103381 


343-77371 


10 


51 


986 


966 


9 




51 


170307 


636690 


815726 


381-97099 


9 


52 


9998 044 


973 


8 




52 


267654 


844984 


50^548506 


429-71757 


8 


53 


101 


979 


7 




53 


365537 


27-056557 


51-303157 


491-10600 


7 


54 


157 


985 


6 




5^ 


464471 


271486 


52-080673 


572-95721 


6 


55 


213 


989 


5 




5£ 


56447S 


48985*d 


88210C 


637-54687 


5 


56 


267 


993 


4 




56 


665565 


< 71174C 


53-708587 


859-43630 


4 


57 


321 


996 


3 




57 


767754 


937233 


54-56130C 


11145-9153 


a 


58 


374 


99S 


2 




5^ 


87106E 


) 28-166425 


55-44151- 


' 1718-8732 


2 


59 


426 


1-0000 OOC 


1 



5i 


) 97552: 


I 399397 


56-35059( 


) 3437-7467 


I 


60 


477 


OOC 


6( 


) 19-08113' 


r 636253 


57-289965 


I Infinite. 







\ 1° 


0° 


' 


3° 


2° 


1 lo 


0° 


' 


1 


NAT 


. COSINE. NAT. COTAN. JJ 



TRAVERSE TABLE 
TO EVERY aUARTER POINT OF THE COMPASS. 



16* 



TlSG Dif. oflat. and dep.for i Point. \Dif. oflat. and dep.for J Pomt. [ T. iv .| j 


dist. 


lat. 


dep. 


dist. 


lat. 


dep. 


dist. 


lat. 


dep. 


dist. 


lot. 


dep. 


1 


01-0 


00-0 


61 


60-9 


03-0 


1 


01-0 


00-1 


61 


60-7 


06-0 


2 


020 


00-1 


62 


61'9 


03-0 


2 


02-0 


00-2 


62 


61-7 


06-1 


3 


03-0 


oo-i 


63 


62-9 


03-1 


3 


03-0 


00-3 


63 


62-7 


06-2 


4 


04-0 


00-2 


64 


63-9 


03-1 


4 


04-0 


00-4 


64 


63-7 


06-3 


5 


05-0 


00-2 


65 


64-9 


03-2 


5 


05-0 


00-5 


65 


64-7 


06-4 


6 


06-0 


00-3 


66 


65-9 


03-2 


6 


06-0 


00-6 


66 


65-7 


06-5 


7 


070 


00-3 


67 


66-9 


03-3 


7 


07-0 


00-7 


67 


66-7 


06-6 


8 


08-0 


00-4 


68 


67-9 


03-3 


8 


08-0 


00-8 


68 


67-7 


06-7 


9 


09-0 


00-4 


69 


68-9 


03-4 


9 


09-0 


00-9 


69 


68-7 


06-8 


10 


100 


00-5 


70 


69-9 


03-4 


10 


10-0 


01-0 


70 


69-7 


06-9 


11 


ll'O 


00-5 


71 


70-9 


03-5 


11 


10-9 


01-1 


71 


70-7 


07-0 


12 


12-0 


00-6 


72 


71-9 


03-5 


12 


11-9 


01-2 


72 


71-7 


07-1 


13 


130 


00-6 


73 


72-9 


03-6 


13 


12-6 


01-3 


73 


72-7 


07-2 


14 


14-0 


00-7 


74 


73-9 


03-6 


14 


13-9 


01-4 


74 


73-6 


07-3 


15 


150 


00-7 


75 


74-9 


03-7 


15 


14-9 


01-5 


75 


74-6 


07-4 


16 


16-0 


00-8 


76 


75-9 


03-7 


16 


15-9 


01-6 


76 


75-6 


07-4 


17 


17-0 


00-8 


77 


76-9 


03-8 


17 


16-9 


01-7 


77 


76-6 


07-5 


18 


18-0 


00-9 


78 


77-9 


03-8 


18 


17-9 


01-8 


78 


77-6 


07-6 


19 


19-0 


00-9 


79 


78-9 


03-9 


19 


18-9 


01-9 


79 


78-6 


07-7 


20 


20-0 


01-0 


80 


79-9 


03-9 


20 


19-9 


02-0 


80 


79-6 


07-8 


21 


21-0 


01-0 


81 


80-9 


04-0 


21 


20-9 


02-1 


81 


80-6 


07-9 


22 


22-0 


01-1 


82 


81-9 


04-0 


22 


21-9 


02-2 


82 


81-6 


OS-0 


23 


23-0 


01-1 


83 


82-9 


04-1 


23 


22-9 


02-3 


83 


82-6 


08-1 


24 


24-0 


01-2 


84 


83-9 


04-1 


24 


23-9 


02-4 


84 


83-6 


08-2 


25 


25-0 


01-2 


85 


84-9 


04-2 


25 


24-9 


02-4 


85 


84-6 


08-3 


26 


26-0 


01-3 


86 


85-9 


04-2 


26 


25-9 


02-5 


86 


85-6 


08-4 


27 


27-0 


01-3 


87 


86-9 


04-3 


27 


26-9 


02-6 


87 


86-6 


08-5 


28 


28-0 


01-4 


88 


87-9 


04-3 


28 


27-9 


02-7 


88 


87-6 


08-6 


29 


29-0 


01-4 


89 


88-9 


04-4 


29 


28-9 


02-8 


89 


88-6 


08-7 


30 


30-0 


01-5 


90 


89-9 


04-4 


30 


29-9 


02-9 


90 


89-6 


08-8 


31 


310 


01-5 


91 


90-9 


04-5 


31 


30-9 


03-0 


91 


90-6 


08-9 


32 


32-0 


01-6 


92 


91-9 


04-5 


32 


31-8 


03-1 


92 


91-6 


09-0 


33 


33-0 


01-6 


93 


92-9 


04-6 


33 


32-8 


03-2 


93 


92-6 


09-1 


34 


34-0 


01-7 


94 


93-9 


04-6 


34 


33-8 


03-3 


94 


93-6 


09-2 


35 


35-0 


01-7 


95 


94-9 


04-7 


35 


34-8 


03-4 


95 


94-5 


09-3 


36 


36-0 


01-8 


96 


95-9 


04-7 


36 


35-8 


03-5 


96 


95-5 


09-4 


' 37 


37-0 


01-8 


97 


96-9 


04-8 


37 


36-8 


03-6 


97 


96-5 


09-5 


1 38 


38-0 


01-9 


98 


97-9 


04-8 


38 


37-8 


03-7 


98 


97-5 


09-6 


39 


39-0 


01-9 


99 


98-9 


04-9 


39 


38-8 


03-8 


99 


98-5 


09-7 


40 


40-0 


02-0 


100 


99-9 


04-9 


40 


39-8 


03-9 


100 


99-5 


09-8 


41 


41-0 


02-0 


101 


100-9 


05-0 


41 


40-8 


04-0 


101 


100-5 


09-9 


42 


41-9 


02-1 


102 


101-9 


05-0 


42 


4r8 


04-1 


102 


101-5 


100 


43 


42-9 


02-1 


103 


102-9 


05-1 


,43 


42-8 


04-2 


103 


102-5 


10-1 


44 


43-9 


02-2 


104 


103-9 


05-1 


44 


43-8 


04-3 


104 


103-5 


10-2 


1 45 


44-9 


02-2 


105 


104-9 


05-2 


45 


44-8 


04-4 


105 


104-5 


10-3 


1 46 


45-9 


02-3 


106 


105-9 


05-2 


46 


45-8 


04-5 


106 


105-5 


10-4 


1 47 


46-9 


02-3 


107 


106-9 


05-3 


47 


46-8 


04-6 


107 


106-5 


10-5 


1 48 


47-9 


02-4 


108 


107-9 


05-3 


48 


47-8 


04-7 


108 


107-5 


10-6 


49 


48-9 


02-4 


109 


108-9 


05-4 


49 


48-8 


04-8 


109 


108-5 


10-7 


50 


49-9 


02-5 


110 


109-9 


• 054 


50 


49-8 


04-9 


110 


109-5 


10-8 


51 


50-9 


02-5 


111 


110-9 


05-5 


51 


50-8 


05-0 


111 


110-5 


10-9 


52 


51-9 


02-6 


112 


111-9 


05-5 


52 


51-7 


05-1 


112 


111-5 


11-0 


53 


52-9 


02-6 


113 


112-9 


05-5 


53 


52-7 


05-2 


113 


112-5 


11-1 


54 


53-9 


02-7 


114 


113-9 


05-6 


54 


53-7 


05-3 


114 


113-5 


n-2 


55 


54-9 


02-7 


115 


114-9 


05-6 


55 


54-7 


05-4 


115 


1I4'5 


11-3 


56 


55-9 


02-8 


116 


115-9 


05-7 


56 


55-7 


05-5 


116 


115-4 


11-4 


57 


56-9 


02-8 


117 


116-9 


05-7 


57 


56-7 


05-6 


117 


116-4 


11-5 


1 58 


57-9 


02-9 


118 


117-9 


05-8 


58 


57-7 


05-7 


118 


117-4 


11-6 


59 


58-9 


02-9 


119 


118-9 


05-8 


59 


58-7 


05-8 


119 


118-4 


11-7 


60 


59-9 


02-9 


120 


119-9 


05-9 


60 


59-7 


05-9 


120 


119-4 


11-8 


dist. 


dep. 


lat. 


dist. 


dep. 


lat. 


dist. 


dep. 


lat. 


dist. 


dep. 


lat. 


For 7 


1 Points. 


For 7^ Points. " |j 



jr. IV.] Dif.oflat. 


and dep. for | Point.] Dif. of lat. and dep. for 


I Point. 187 ] 


(list. 


lat. 


dep. 


dist. 


lat. 


dep. 


dist. 


lat. 


dep. 


dist. 


lat. 


dep. 


1 


oi-o 


00-1 


61 


60-3 


08-9 


1 


01-0 


00-2 


61 


59-8 


11-9 


2 


02-0 


00-3 


62 


61-3 


09-1 


2 


020 


00-4 


62 


60-8 


12-1 


3 


03-0 


00-4 


63 


62-3 


09-2 


3 


02-9 


00-6 


63 


61-8 


12-3 


4 


04-0 


00-6 


64 


63-3 


09-4 


4 


03-9 


00-8 


64 


62-8 


12-5 


5 


04-9 


00 7 


65 


64-3 


09-5 


5 


04-9 


010 


65 


63-7 


12-7 


6 


05-9 


00-9 


66 


65-3 


09-7 


6 


05-9 


01-2 


66 


64-7 


12-9 


7 


06-9 


01-0 


67 


66-3 


09-8 


7 


06-9 


01-4 


67 


65-7 


13-1 


8 


07-9 


01-2 


68 


67-3 


10-0 


8 


07-8 


01-6 


68 


66-7 


13-3 


9 


08-9 


01-3 


69 


68-2 


10-1 


9 


08-8 


01-8 


69 


67-7 


13-5 


10 


09-9 


01-5 


70 


69-2 


10-3 


10 


09-8 


02-0 


70 


68-7 


13-7 


11 


10-9 


01-6 


71 


70-2 


10-4 


11 


10-8 


02-1 


71 


69-6 


13-9 


12 


11-9 


01-8 


72 


71-2 


10-6 


12 


11-8 


02-3 


72 


70-6 


140 


13 


12-9 


01-9 


73 


72-2 


10-7 


13 


12-7 


02-5 


73 


71-6 


14-2 


14 


13-8 


02-1 


74 


73-2 


10-9 


14 


13-7 


02-7 


74 


72-6 


14-4 


15 


14-8 


02-2 


75 


74-2 


11-0 • 


15 


14-7 


02-9 


75 


73-6 


14-6 


16 


15-8 


02-3 


76 


75-2 


ll-l 


16 


15-7 


03-1 


76 


74-5 


14-8 


17 


16-8 


02-5 


77 


76-2 


11-3 


17 


16-7 


03-3 


77 


75-5 


15-0 


18 


17-8 


02-6 


78 


77-2 


11-4 


18 


17-7 


03-5 


78 


76-5 


15-2 


19 


18-8 


02-8 


79 


78-1 


11-6 


19 


18-6 


03-7 


79 


77-5 


15-4 


20 


19-8 


02-9 


80 


79-1 


11-7 


20 


19-6 


03-9 


80 


78-5 


156 


21 


20-8 


03-1 


81 


80-1 


11-9 


21 


20-6 


04-1 


81 


79-4 


15-8 


22 


21-8 


03-2 


82 


81-1 


120 


22 


21-6 


04-3 


82 


80-4 


16-0 


23 


22-7 


03-4 


83 


82-1 


12-2 


23 


22-6 


04-5 


83 


81-4 


16-2 


1 24 


23-7 


03-5 


84 


83-1 


12-3 


24 


23-5 


04-7 


84 


82-4 


16-4 


1 25 


24-7 


03-7 


85 


84-1 


12-5 


25 


24-5 


04-9 


85 


83-4 


16-6 


26 


25-7 


03-8 


86 


85-1 


12-6 


26 


25-5 


05-1 


86 


84-3 


16-8 


27 


26-7 


04 


87 


86-1 


12-8 


27 


26-5 


05-3 


87 


85-3 


17-0 


28 


27-7 


04-1 


88 


87-0 


12-9 


28 


27-5 


05-5 


88 


86-3 


17-2 


29 


28-7 


04-3 


89 


88-0 


13-1 


29 


28-4 


05-7 


89 


87-3 


17-4 


30 


29-7 


04 4 


90 


89-0 


13-2 


30 


29-4 


05-9 


90 


88-3 


17-6 


31 


30-7 


04-5 


91 


90-0 


13-3 


31 


30-4 


06-0 


91 


89-2 


17-8 


32 


31-7 


04-7 


92 


91-0 


13-5 


32 


31-4 


06-2 


92 


90-2 


18-0 


33 


32-6 


04-8 


93 


92-0 


13-6 


33 


32-4 


06-4 


93 


91-2 


18-1 


34 


33-6 


05-0 


94 


93-0 


13-8 


34 


33-3 


06-6 


94 


92-2 


18-3 


35 


34-6 


05-1 


95 


94-0 


13-9 


35 


34-3 


06-8 


95 


93-2 


18-5 


36 


35-6 


05-3 


96 


95-0 


14-1 


36 


35-3 


07-0 


96 


94-2 


18-7 


37 


36-6 


05-4 


97 


95-9 


14-2 


37 


36-3 


07-2 


97 


95-1 


18-9 


38 


37-6 


05-6 


98 


96-9 


14-4 


33 


37-3 


07-4 


98 


96-1 


19-1 


39 


38-6 


05-7 


99 


97-9 


14-5 


39 


38-2 


07-6 


90 


97-1 


19-3 


40 


39-6 


05-9 


100 


98-9 


14-7 


40 


39-2 


07-8 


100 


981 


19-5 


41 


40-6 


06-0 


101 


99-9 


14-8 


41 


40-2 


08-0 


101 


99-1 


19-7 


42 


41-5 


06-2 


102 


100-9 


15-0 


42 


41-2 


08-2 


102 


100-0 


19-9 


43 


42-5 


06-3 


103 


101-9 


15-1 


43 


42-2 


08-4 


103 


101-0 


20-1 


44 


43-5 


06-5 


104 


102-9 


15-3 


44 


43-2 


08-6 


104 


102-0 


20-3 


45 


44-5 


06-6 


105 


103-9 


15-4 


45 


44-1 


08-8 


105 


103-0 


20-5 


46 


45-5 


06-7 


106 


104-8 


15-5 


46 


45-1 


09-0 


106 


104-0 


20-7 


47 


46-5 


06-9 


107 


105-8 


15-7 


47 


46-1 


09-2 


107 


104-9 


20-9 


48 


47-5 


07-0 


108 


106-8 


15-8 


48 


47-1 


09-4 


108 


105-9 


21-1 


49 


48-5 


07-2 


109 


107-8 


16-0 


49 


48-1 


09-6 


109 


106-9 


21-3 


50 


49-5 


07-3 


110 


108-8 


16-1 


50 


49-0 


09-8 


110 


107-9 


21-5 


51 


50-4 


07-5 


HI 


109-8 


16-3 


51 


50-0 


10-0 


111 


108-9 


21-7 


52 


51-4 


07-6 


112 


110-8 


16-4 


52 


51-0 


10-1 


112 


109-8 


21-9 


53 


52-4 


07-8 


113 


111-8 


16-6 


53 


52-0 


10-3 


113 


110-8 


22-0 


54 


53-4 


07-9 


114 


112-8 


16-7 


54 


53-0 


10-5 


114 


111-8 


22-2 


55 


54-4 


08-1 


115 


113-7 


16-9 


55 


53-9 


10-7 


115 


112-8 


22-4 


56 


55-4 


08-2 


116 


114-7 


17-0 


56 


54-9 


10-9 


116 


113-8 


22-6 


57 


56-4 


08-4 


117 


115-7 


17-2 


57 


55-9 


11-1 


117 


114-7 


22-8 


58 


57-4 


08-5 


118 


116-7 


17-3 


58 


56-9 


11-3 


118 


115-7 


23-0 


59 


58-4 


08-7 


119 


117-4 


17-5 


59 


57-9 


11-5 


119 


116-7 


23-2 


60 


59-3 


08-8 


120 


118-7 


17-6 


60 


58-8 


11-7 


120 


117-7 


23-4 


dist. 


, dep. 


lat. 


dist. 


dep. 


lat. 


dist. 


dep. 


lat. 


dist. 


dep. 


lat. 


I For 


7iPo 


ints. 


For' 


7 Foil 


Its. II 



188 Dif. oflat. <^ dep.for U Point\Dif. oflat. d^ dep.for U Point. [ T. i v. I| 




dist. 


lat. 


dep. 


dist. 


lat. 


dep. 


dist. 


lat. 


dep. 


dist. 


lat. 


dep. j 




1 1 


01-0 


00-2 


61 


59-2 


14-8 


1 


01-0 


00-3 


61 


58-4 


17-7 




2 


01-9 


00-5 


62 


60-1 


15-1 


2 


Ci-9 


00-6 


62 


59-3 


13-0 




3 


02-9 


00-7 


63 


61-1 


15-3 


3 


02-9 


00-9 


63 


60-3 


18-3 




4 


03-9 


01-0 


64 


62-1 


15-6 


4 


03-8 


01-2 


64 


61-2 


18-6 




5 


04-9 


01-2 


65 


63-1 


15-8 


5 


04-8 


01-5 


65 


62-2 


18-9 




6 


05-8 


01-5 


66 


64-0 


16-0 


6 


05-7 


01-7 


66 


63-2 


19-2 




7 


06-8 


01-7 


67 


65-0 


16-3 


7 


06-7 


02-0 


67 


641 


19-4 




8 


07-8 


01-9 


68 


66-0 


16-5 


8 


07-7 


02-3 


68 


65-1 


19-7 




9 


08-7 


02-2 


69 


66-9 


16-8 


9 


03-6 


02-6 


69 


660 


20-0 




10 


09-7 


02-4 


70 


67-9 


17-0 


10 


09-6 


02-9 


70 


67-0 


20-3 




11 


10-7 


02-7 


71 


68-9 


17-3 


11 


10-5 


03-2 


71 


67-9 


20-6 




12 


11-6 


02-9 


72 


69-8 


17-5 


12 


11-5 


03-5 


72 


68-9 


20-9 




13 


12-6 


03-2 


73 


70-8 


17-7 


13 


12-4 


03-3 


73 


69-9 


21-2 




14 


13-6 


03-4 


74 


71-8 


18-0 


14 


13-4 


04-1 


74 


70-8 


21-5 




15 


14-6 


03-6 


75 


72-8 


18-2 


15 


14-4 


04-4 


75 


71-8 


21-8 




16 


15-5 


03-9 


76 


73-7 


18-5 


16 


15-3 


04-6 


76 


72-7 


22-1 




17 


16-5 


04-1 


77 


74-7 


18-7 


17 


163 


04-9 


77 


73-7 


22-3 




18 


17-5 


04-4 


78 


75-7 


19-0 


18 


17-2 


05-2 


78 


74-6 


22-6 




19 


18-4 


04-6 


79 


76-6 


19-2 


19 


13-2 


05-5 


79 


75-6 


22-9 




20 


19-4 


04-9 


80 


77-6 


19-4 


20 


19-1 


05-8 


80 


76-6 


23-2 




21 


20-4 


05-1 


81 


78-6 


19-7 


21 


20-1 


06-1 


81 


77-5 


23-5 




22 


21-3 


05-3 


82 


79-5 


19-9 


22^ 


21-1 


06-4 


82 


78-5 


23-3 




23 


22-3 


05-6 


83 


80-5 


20-2 


23 


22-0 


06-7 


83 


79-4 


24-1 




24 


23-3 


05-8 


84 


81-5 


20-4 


24 


23-0 


07-0 


84 


80-4 


24-4 




25 


24-3 


06-1 


85 


82-5 


20-7 


25 


23-9 


07-3 


85 


81-3 


24-7 




26 


25-2 


06-3 


86 


83-4 


20-9 


26 


24-9 


07-5 


86 


82-3 


25-0 




27 


26'2 


06-6 


87 


84-4 


21-1 


27 


25-8 


07-3 


87 


83-3 


25-2 




28 


27'2 


06-8 


83 


85-4 


21-4 


28 


26-8 


03-1 


88 


84-2 


25-5 




29 


28-1 


07'0 


89 


86-3 


21-6 


29 


27-8 


03-4 


89 


85-2 


25-8 




30 


29-1 


07-3 


90 


87-3 


21-9 


30 


28-7 


08-7 


90 


86-1 


261 




31 


30-1 


07-5 


91 


88-3 


22-1 


31 


29-7 


09-0 


91 


87-1 


26-4 




32 


31-0 


07-8 


92 


89-2 


22-4 


32 


30-6 


09-3 


92 


88-0 


26-7 




33 


32-0 


08'0 


93 


90-2 


22-6 


33 


31-6 


09-6 


93 


89-0 


27-0 




34 


33-0 


08-3 


94 


91-2 


22-8 


34 


32-5 


09-9 


94 


90-0 


27-3 




35 


34-0 


08-5 


95 


92-2 


23-1 


35 


33-5 


10-2 


95 


90-9 


27-6 




36 


34-9 


08-7 


96 


93-1 


23-3 


36 


34-5 


10-4 


96 


91-9 


27-9 




37 


35-9 


09-0 


97 


94-1 


23-6 


37 


35-4 


10-7 


97 


92-8 


28-2 




38 


36-9 


09-2 


93 


95-1 


23-3 


38 


36-4 


11-0 


93 


93-3 


28-4 




39 


37-8 


09-5 


99 


96-0 


24-1 


39 


37-3 


11-3 


99 


94-7 


28-7 




40 


38-8 


09-7 


100 


97-0 


24-3 


40 


38-3 


11-6 


100 


95-7 


29-0 




41 


39-8 


10-0 


101 


93-0 


24-5 


41 


39-2 


11-9 


101 


96-7 


29-3 




42 


40-7 


10-2 


102 


93-9 


24-3 


42 


40-2 


12-2 


102 


97-6 


29-6 




43 


41-7 


10-4 


103 


99-9 


25-0 


43 


41-2 


12-5 


103 


98-6 


29-9 




44 


42-7 


10-7 


104 


100-9 


25-3 


44 


42-1 


12-G 


104 


99-5 


30-2 




45 


43-7 


10-9 


105 


101-9 


25-5 


45 


43-1 


13-1 


105 


100-5 


30-5 




46 


44-6 


11-2 


106 


102-3 


25-3 


46 


44-0 


13-3 


106 


101-4 


30-8 




47 


45-6 


11-4 


107 


103-8 


26-0 


47 


45-0 


13-e 


107 


102-4 


31-1 




48 


46-6 


11-7 


108 


104-3 


26-2 


48 


45-9 


13-c 


103 


103-4 


31-4 




49 


47-5 


11-9 


109 


105-7 


26-5 


49 


46-9 


14-'. 


109 


104-3 


31-6 




50 


48-5 


12-2 


110 


106-7 


26-7 


50 


47-9 


14-£ 


) 110 


105-3 


31-9 




51 


49-5 


12-4 


HI 


107-7 


27-0 


51 


48-8 


14-{ 


HI 


106-2 


32-2 




52 


50-4 


12-6 


112 


108-6 


27-2 


52 


49-8 


15- 


112 


107-2 


32-5 




53 


51-4 


12-9 


113 


109-6 


27-5 


53 


50-7 


15-^ 


I 113 


108-1 


32-8 




54 


52-4 


131 


114 


110-6 


27-7 


54 


51-7 


15-" 


r 114 


109-1 


331 




55 


53-4 


13-4 


115 


111-6 


27-9 


55 


52-6 


16-( 


3 115 


110-1 


33-4 




56 


54-3 


13-6 


116 


112-5 


28-2 


56 


53-6 


16- 


J 116 


111-0 


33-7 




57 


55-3 


13-9 


117 


113-5 


28-4 


57 


54-5 


16- 


5 117 


112-0 


340 




58 


56-3 


141 


118 


114-5 


28-7 


58 


55-5 


16- 


3 118 


112-9 


34-2 




59 


57-2 


14-3 


119 


115-4 


28-9 


59 


56-5 


17- 


1 119 


113-9 


34-5 




60 


58-2 


14-6 120 
lat. dist 


116-4 


29-2 


60 


57-4 


17- 


i 120 


114-8 


34-8 




dis 


t. dep. 


dep. 


lat. 


\dist 


. dep. 


lat 


dist 


, dep. 


lat. 




11 For 6S Pc 


)ints. 


For 6J Points. 





\t. IV.] Dif. oflat. tf- dep.for 1| Point. Dif. oflat. <^ dep.J 


or 2 Point. 189 , 




dist. 


lut. 


dep. 


dist. 


lat. 


dep 


dist. 


lat. 


dep. 


dist. 


lat. 


dep. 




\ 


00-9 


00-3 


61 


5T4. 


20-5 


1 


00-9 


00-4 


61 


56-4 


23-3 




2 


01-9 


00-7 


62 


58-4 


20-9 


2 


01-8 


00-8 


62 


57-3 


23-7 




3 


02-8 


01-0 


63 


59-3 


21-2 


3 


02-8 


01-1 


63 


58-2 


24-1 




4 


03-8 


01-3 


64 


60-3 


21-6 


4 


03-7 


01-5 


64 


59-1 


24-5 




5 


04-7 


01-7 


65 


61-2 


21-9 


5 


04-6 


01-9 


65 


60-1 


24-9 




6 


05-6 


02 


66 


62-1 


22-2 


6 


05-5 


02-3 


66 


61-0 


25-3 




7 


06-6 


02-4 


67 


63-1 


22-6 


7 


06-5 


02-7 


67 


61-9 


25-6 




8 


07-5 


02-7 


68 


64-0 


22-9 


8 


07-4 


03-1 


68 


62-8 


26-0 




9 


08-5 


03-0 


69 


65-0 


23-2 


9 


08-3 


03-4 


69 


63-8 


26-4 




10 


09-4 


03-4 


70 


65-9 


23-6 


10 


09-2 


03-8 


70 


64-7 


26-8 




11 


10-4 


03-7 


71 


66-8 


23-9 


11 


10-2 


04-2 


71 


65-6 


27-2 




12 


11-3 


04-0 


72 


67-8 


24-3 


12 


111 


04-6 


72 


66-5 


27-6 




13 


12-2 


04-4 


73 


68-7 


24-6 


13 


12-0 


05-0 


73 


67-4 


27-9 




14 


13-2 


04-7 


74 


69-7 


24-9 


14 


12-9 


05-4 


74 


68-4 


28-3 




15 


14-1 


05-1 


75 


70-6 


25-3 


15 


13-9 


05-7 


75 


69-3 


28-7 




16 


15-1 


05-4 


76 


71-6 


25-6 


16 


14-8 


06-1 


76 


70-2 


29-1 




17 


16-0 


05-7 


77 


72-5 


25-9 


17 


15-7 


06-5 


77 


71-1 


29-5 




18 


170 


06-1 


78 


73-4 


26-3 


18 


16-6 


06-9 


78 


72-1 


29-9 




19 


17-9 


06-4 


79 


74-4 


26-6 


19 


17-6 


07-3 


79 


73-0 


30-2 




20 


18-8 


06-7 


80 


75-3 


26-9 


20 


18-5 


07-7 


80 


73-9 


30-6 




21 


19-8 


07-1 


81 


76-3 


27-3 


21 


19-4 


08-0 


81 


74-8 


31-0 




22 


20-7 


07-4 


82 


77-2 


27-6 


22 


20-3 


08-4 


82 


75-8 


31-4 




23 


21-7 


07-7 


83 


78-1 


28-0 


23 


21-3 


08-8 


83 


76-7 


31-8 




24 


22-6 


08-1 


84 


79-1 


28-3 


24 


22-2 


09-2 


84 


77-6 


32-1 




25 


23-5 


08-4 


85 


80-0 


28-6 


25 


23-1 


09-6 


85 


78-5 


32-5 




26 


24-5 


08-8 


86 


81-0 


29-0 


26 


24-0 


10-0 


86 


79-5 


32-9 




27 


25-4 


09-1 


87 


81-9 


29-3 


27 


24-9 


10-3 


87 


60-4 


33-3 




28 


26-4 


09-4 


88 


82-9 


29-6 


28 


25-9 


10-7 


88 


81-3 


33-7 




29 


27-3 


09-8 


89 


83-8 


30-0 


29 


26-8 


11-1 


89 


82-2 


34-1 




30 


28-2 


10-1. 


90 


84-7 


30-3 


30 


27-7 


11-5 


90 


83-2 


34-4 




31 


29-2 


10-4 


91 


85-7 


30-7 


31 


28-6 


11-9 


91 


84-1 


34-8 




32 


30-1 


10-8 


92 


86-6 


31-0 


32 


29-6 


12-2 


92 


85-0 


35-2 




33 


31-1 


11-1 


93 


87-6 


31-3 


33 


30-5 


12-6 


93 


85-9 


35-6 




34 


320 


11-5 


94 


88-5 


31-7 


34 


31-4 


13-0 


94 


86-8 


36-0 




35 


330 


11-8 


95 


89-4 


32-0 


35 


32-3 


13-4 


95 


87-8 


36-4 




36 


33-9 


12-1, 


96 


90-4 


32-3 


36 


33-3 


13-3 


96 


88-7 


36-7 




37 


34-8 


12-5 


97 


91-3 


32-7 


37 


34-2 


14-2 


97 


89-6 


371 




38 


35-8 


12-8 


98 


92-3 


33-0 


38 


35-1 


14-5 


98 


90-5 


37-5 




39 


36-7 


13-1 


99 


93-2 


33-3 


39 


36-0 


14-9 


99 


91-5 


37-9 




40 


37-7 


13-5 


100 


94-2 


33-7 


40 


37-0 


15-3 


100 


92-4 


38-3 




41 


38-6 


13-8 


101 


95-1 


34-0 


41 


37-9 


15-7 


101 


93-3 


38-7 




42 


39-5 


14-1 


102 


96-0 


34-4 


42 


38-8 


16-1 


102 


94-2 


39-0 




43 


40-5 


14-5 


103 


97-0 


34-7 


43 


39-7 


16-5 


103 


95-2 


39-4 




44 


41-4 


14-8 


104 


97-9 


35-0 


44 


40-6 


16-8 


104 


96-1 


39-8 




45 


42-4 


15-2 


105 


98-9 


35-4 


45 


41-6 


17-2 


105 


97-0 


40-2 




46 


43-3 


15-5 


106 


99-8 


35-7 


46 


42-5 


17-6 


106 


97-9 


40-6 




47 


44-3 


15-8 


107 


100-7 


36-0 


47 


43-4 


18-0 


107 


98-9 


41-0 




48 


45-2 


16-2 


108 


101-7 


36-4 


48 


44-4 


18-4 


108 


99-8 


41-3 




49 


46-1 


16-5 


109 


102-6 


36-7 


49 


45-3 


18-8 


109 


100-7 


41-7 




50 


47-1 


16-8 


110 


103-6 


37-1 


50 


46-2 


19-1 


110 


101-6 


42-1 




51 


48-0 


17-2 


HI 


104-5 


37-4 


51 


47-1 


19-5 


111 


102-6 


42-5 




52 


49-0 


17-5 


112 


105-4 


37-7 


52 


48-0 


19-9 


112 


103-5 


42-9 




53 


49-9 


17-9 


113 


106-4 


381 


53 


49-0 


20-3 


113 


104-4 


43-2 




54 


50-8 


18-2 


114 


107-3 


38-4 


54 


49-9 


20-7 


114 


105-3 


43-6 




55 


51-8 


18-5 


115 


108-3 


38-7 


55 


50-8 


21-0 


115 


106-3 


440 




56 


52-7 


18-9 


116 


109-2 


39-1 


56 


51-7 


21-4 


116 


107-2 


44-4 




57 


53-7 


19-2 


117 


110-2 


39-4 


57 


52-7 


21-8 


117 


108-1 


44-8 




58 


54-6 


19-5 


118 


111-1 


39-7 


58 


53-6 


22-2 


118 


109-0 


45-2 




59 


55-5 


19-9 


119 


1120 


40-1 


59 


54-5 


22-6 


119 


109-9 


45-5 




60 


56-5 


20-2 


120 


113-0 


40-4 


60 


55-4 


230 


120 


110-9 


45-9 




dist. 


dep. 


Ut. 


dist. 


dep. 


lat. 


dist. 


dep. 


lat. 


dist. 


dep. 


lat. 




For 6i Poi 


tits. 


' For 6 


Poin 


ts. 1 





190 Dif. oflat. (f- dep.for 2| Point. \Dif. oflat. <^ dep.f 


or ^^ Point. [T. iv,| 


dist. 


lat. 


dep. 


dist. 


lat. 


dep. 


dist. 


lat. 


dep. 


dist. 


lat. 


dep. 


1 


00-9 


00-4 


61 


55-1 


26-1 


1 


00-9 


00-5 


61 


53-8 


28-8 


2 


01-8 


00-9 


62 


56-0 


26-5 


2 


01-8 


00-9 


62 


54-7 


29-2 


3 


02-7 


01-3 


63 


57-0 


26-9 


3 


02-6 


01-0 


63 


55-6 


29-7 


4 


03-6 


01-7 


64 


57-9 


27-4 


4 


03-5 


01-9 


64 


56-4 


30-2 


5 


04-5 


02-1 


65 


58-3 


27-8 


5 


04-4 


02-4 


65 


57-3 


30-6 


6 


05-4 


02-6 


66 


59-7 


28-2 


6 


05-3 


02-8 


60 


58-2 


31-1 


7 


06-3 


03-0 


67 


00-6 


28-6 


7 


06-2 


03-3 


67 


59-1 


31-6 


8 


07-2 


03-4 


68 


61-5 


29-1 


8 


07-1 


03-8 


68 


60-0 


32-1 


9 


08-1 


03-8 


69 


62-4 


29-5 


9 


07-9 


04-2 


69 


60-9 


32-5 


10 


09-0 


04-3 


70 


63-3 


29-9 


10 


08-8 


04-7 


70 


61-7 


33-0 


11 


09-9 


04-7 


71 


64-2 


30-4 


11 


09-7 


05-2 


71 


62-6 


33-5 


12 


10-8 


05-1 


72 


65-1 


30-8 


12 


10-6 


05-7 


72 


63-5 


33-9 


13 


11-8 


05-6 


73 


66-0 


31-2 


13 


11-5 


06-1 


73 


64-4 


34-4 


14 


12-7 


06-0 


74 


66-9 


31-6 


14 


12-3 


06-6 


74 


65-3 


34-9 


15 


13-6 


06'4 


75 


67-8 


32-1 


15 


13-2 


07-1 


. 75 


66-1 


35-4 


16 


14-5 


06-8 


76 


68-7 


32-5 


16 


14-1 


07-5 


76 


67-0 


35-8 


17 


15-4 


07-3 


77 


69-6 


32-9 


17 


15-0 


08-0 


77 


67-9 


36-3 


18 


16-3 


07-7 


78 


70-5 


33-4 


18 


15-9 


08-5 


78 


68-8 


36-8 


19 


17-2 


08-1 


79 


71-4 


33-8 


19 


16-8 


09-0 


79 


69-7 


37-2 


20 


181 


08-6 


80 


72-3 


34-2 


20 


17-6 


09-4 


80 


70-6 


37-7 


21 


19-0 


09-0 


81 


73-2 


34-6 


21 


18-5 


09-9 


81 


71-4 


38-2 


22 


19-9 


09-4 


82 


74-1 


35-1 


22 


19-4 


10-4 


82 


72-3 


38-a 


23 


20-8 


09-8 


83 


75-0 


35-5 


23 


20-3 


10-8 


83 


73-2 


39-1 


24 


21-7 


10-3 


84 


75-9 


35-9 


24 


21-2 


11-3 


84 


74-1 


39-6 


25 


22-6 


10-7 


85 


76-8 


36-3 


25 


22-1 


11-8 


85 


75-0 


40-1 


26 


23-5 


11-1 


86 


77-7 


36-8 


26 


22-9 


12-3 


86 


75-9 


40-5 


27 


24-4 


11-5 


87 


78-6 


37-2 


27 


23-8 


12-7 


87 


76-7 


41-0 


28 


25-3 


12-0 


88 


79-6 


37-6 


28 


24-7 


13-2 


88 


77-6 


41-5 


29 


26-2 


12-4 


89 


80-5 


38-1 


29 


25-6 


13-7 


89 


78-5 


41*9 


30 


27-1 


12-8 


90 


81-4 


38-5 


30 


26-5 


14-1 


90 


79-4 


42-4 


31 


28-0 


13'3 


91 


82-3 


38-9 


31 


27-3 


14-6 


91 


80-3 


42-9 


32 


28-9 


13-7 


92 


83-2 


39-3 


32 


28-2 


15-1 


92 


81-1 


43-4 


33 


29-8 


14-1 


93 


84-1 


39-8 


33 


29-1 


15-6 


93 


82-0 


43-8 


34 


30-7 


14-5 


94 


85-0 


40-2 


34 


30-0 


16-0 


94 


82-9 


44-3 


35 


31-6 


15-0 


95 


85-9 


40-6 


35 


30-9 


16-5 


95 


83-8 


44-8 


36 


32-5 


15-4 


96 


86-8 


41-1 


36 


31-8 


17-0 


96 


84-7 


45-2 


37 


33-4 


15-8 


97 


87-7 


41-5 


37 


32-6 


17-4 


97 


85-6 


45-7 


38 


34-4 


16-2 


98 


88-6 


41-9 


38 


33-5 


17-9 


98 


86-4 


46-2 


39 


35-3 


16-7 


99 


89-5 


42-3 


39 


34-4 


18-4 


99 


87-3 


46-7 


40 


36-2 


17-1 


100 


90-4 


42-8 


40 


35-3 


18-9 


100 


88-2 


47-1 


41 


37-1 


17-5 


101 


91-3 


43-2 


.41 


36-2 


19-3 


101 


89-1 


47-6 


42 


38-0 


18-0 


102 


92-2 


43-6 


42 


37-0 


19-8 


102 


90-0 


48-1 


43 


38-9 


18-4 


103 


93-1 


44-0 


43 


37-9 


20-3 


103 


90-8 


48-5 


44 


39-8 


18-8 


104 


94-0 


44-5 


44 


38-8 


20-7 


104 


91-7 


49-0 


45 


40-7 


19-2 


105 


94-9 


44-9 


45 


39-7 


21-2 


105 


92-6 


49-5 


46 


41-6 


19-7 


106 


95-8 


45-3 


46 


40-6 


21-7 


106 


93-5 


50-0 


47 


42-5 


20-1 


107 


96-7 


45-8 


47 


41-5 


22-2 


107 


94-4 


50-4 


48 


43-4 


20-5 


108 


97-6 


46-2 


48 


42-3 


22-6 


108 


95-3 


50-9 


49 


44-3 


21-0 


109 


98-5 


46-6 


49 


43-2 


23-1 


109 


96-1 


51-4 


50 


45-2 


21-4 


110 


99-4 


47-0 


50 


44-1 


23-6 


110 


97-0 


51-8 


51 


46-1 


21-8 


111 


100-3 


47-5 


51 


45-0 


24-0 


111 


97-9 


52-3 


52 


47-0 


22-2 


112 


101-2 


47-9 


52 


45-9 


24-5 


112 


98-8 


52-8 


53 


47-9 


22-7 


113 


102-1 


48-3 


53 


467 


25-0 


113 


99-7 


53-3 


54 


48-8 


23-1 


114 


103-1 


48-7 


54 


47-6 


25-5 


114 


100-5 


53-7 


55 


49-7 


23-5 


115 


104-0 


49-2 


55 


48-5 


25-9 


115 


101-4 


54-2 


56 


50-6 


23-9 


116 


104-9 


49-6 


56 


49-4 


26-4 


116 


102-3 


54-7 


57 


51-5 


24-4 


117 


105-8 


50-0 


57 


50-3 


26-9 


117 


103-2 


55-1 


58 


52-4 


24-8 


118 


106-7 


50-5 


58 


51-2 


27-3 


118 


104-1 


55-6 


59 


53-3 


25-2 


119 


107-6 


50-9 


59 


52-0 


27-8 


119 


105-0 


56-1 


60 


54-2 


25-7 


120 


108-5 


51-3 


60 


52-9 


28-3 


120 


105-8 


56-6 


dist. 


dep. 


lat. [dist. 


dep. 


lat. 


dist. 


dep. 


lat. 


dist. 


dep. 


lat. 


I 


For 51 Foil 


us. 


~ FOTbh 


Poin 


II 



T. iv.J Dif. of lat. d^ dep. for ^\ Points. 


Dif of lat. 4- dep. for 3 Points. 191 {] 


dlst. 


lat. 


dep. 


dist. 


lat. 


dep. 


dist. 


Lat. 


dep. 


dist. 


lat. 


dep. 


1 


00-0 


00-5 


61 


52-3 


31-4 


1 


00-8 


00-6 


61 


50-7 


33-9 


2 


01-7 


01-0 


62 


53-2 


31-9 


2 


01-7 


00-1 


62 


51-5 


34-4 


3 


02-6 


01-5 


63 


54-0 


32-4 


3 


02-5 


01-7 


63 


52-4 


35-0 


4 


03-4 


02-1 


64 


54-9 


32-9 


4 


03-3 


02-2 


64 


53-2 


35-6 


5 


04-3 


02-6 


65 


55-8 


33-4 


5 


04-2 


02-8 


65 


54-0 


36-1 


6 


05-1 


03-1 


66 


56-6 


33-9 


6 


05-0 


03-3 


66 


54-9 


36-7 


7 


06-0 


03-6 


67 


57-5 


34-4 


7 


05-8 


03-9 


67 


55-7 


37-2 


8 


06-9 


04-1 


68 


58-3 


35-0 


8 


06-7 


04-4 


68 


56-5 


37-8 


9 


07-7 


04-6 


69 


59-2 


35-5 


9 


07-5 


05-0 


69 


57-4 


38-3 


10 


08-6 


05-1 


70 


60-0 


360 


10 


08-3 


05-6 


70 


58-2 


38-9 


11 


09-4 


05-7 


71 


60-9 


36-5 


11 


09-1 


06-1 


71 


59-0 


39-4 


12 


10-3 


08-2 


72 


61-8 


37-0 


12 


100 


06-7 


72 


59-9 


40-0 


13 


11-2 


06-7 


73 


62-6 


37-5 


13 


10-8 


07-2 


73 


60-7 


40-6 


14 


12-0 


07-2 


74 


63-5 


33-0 


14 


11-6 


07-8 


74 


61-5 


41-1 


15 


12-9 


07-7 


75 


64-3 


38-6 


15 


12-5 


08-3 


75 


62-4 


41-7 


16 


13-7 


08-2 


76 


65-2 


39-1 


16 


13-3 


08-9 


76 


63-2 


42-2 


17 


14-6 


08-7 


77 


66-0 


39-6 


17 


14-1 


09-4 


77 


64-0 


42-8 


18 


15-4 


09-3 


78 


66-9 


40-1 


18 


15-0 


10-0 


78 


64-8 


43-3 


19 


16-3 


09-3 


79 


67-8 


40-6 


19 


15-8 


10-6 


79 


65-7 


43-9 


20 


17-2 


10-3 


80 


68-6 


41-1 


20 


16-6 


11-1 


80 


66-5 


44-4 


21 


18-0 


10-8 


81 


69-5 


41-6 


21 


17-5 


11-7 


81 


67-3 


45'0 


22 


18-9 


11-3 


82 


70-3 


42-2 


22 


18-3 


12-2 


82 


68-2 


45-6 


23 


19-7 


11-8 


83 


71-2 


42-7 


23 


19-1 


12-8 


83 


69-0 


46-1 


24 


20-6 


12-3 


84 


72-0 


43-2 


24 


20-0 


13-3 


84 


69-8 


46-7 


25 


21-4 


12-9 


85 


72-9 


43-7 


25 


20-8 


13-9 


85 


70-7 


47-2 


26 


22-3 


13-4 


86 


73-8 


44-2 


26 


21-6 


14-4 


86 


71-5 


47-8 


27 


23-2 


13-9 


87 


74-6 


44-7 


27 


22-4 


15-0 


87 


72-3 


48-3 


28 


24 


14-4 


88 


75-5 


45-2 


28 


23-3 


15-6 


88 


73-2 


48-9 


29 


24-9 


14-9 


89 


76-3 


45-7 


29 


24-1 


16-1 


89 


74-0 


49-4 


30 


25-7 


15-4 


90 


77-2 


46-3 


30 


24-9 


16-7 


90 


74-8 


50-0 


31 


26-6 


15-9 


91 


78-1 


46-8 


31 


25-8 


17-2 


91 


75-7 


50-6 


32 


27-4 


16-4 


92 


78-9 


47-3 


32 


26-6 


17-8 


92 


76-5 


511 


33 


28-3 


17-0 


93 


79-8 


47-8 


33 


27-4 


18-3 


93 


77-3 


51-7 


34 


29-2 


17-5 


94 


80-6 


48-3 


34 


28-3 


18-9 


94 


78-2 


52-2 


35 


30-0 


18-0 


95 


81-5 


48-8 


35 


29-1 


19-4 


95 


79-0 


52-8 


36 


30-9 


18-5 


96 


82-3 


49-3 


36 


29-9 


20-0 


96 


79-8 


53-3 


37 


31-7 


19-0 


97 


83-2 


49-9 


37 


30-8 


20-6 


97 


80-6' 


53-9 


38 


32-6 


19-5 


98 


84-1 


50-4 


38 


31-6 


2M 


98 


81-5 


54-4 


39 


33-5 


20-0 


99 


84-9 


50-9 


39 


32-4 


21-7 


99 


82-3 


55-0 


40 


34-3 


20-6 


100 


85-8 


51-5 


40 


33-3 


22-2 


100 


83-1 


55-6 


41 


35-2 


21-1 


101 


86-6 


51-9 


41 


34-1 


22-8 


101 


84-0 


56-1 


42 


36-0 


21-6 


102 


87-5 


52-4 


42 


34-9 


23-3 


102 


84-8 


56-7 


43 


36-9 


22-1 


103 


88-3 


52-9 


43 


35-8 


23-9 


103 


85-6 


57-2 


44 


37-7 


22-6 


104 


89-2 


53-5 


44 


36-6 


24-4 


104 


86-5 


57-8 


45 


38-6 


23-1 


105 


90-1 


54-0 


45 


37-4 


25-0 


105 


87-3 


58-3 


46 


39-5 


23-6 


106 


90-9 


54-5 


46 


38-2 


25-6 


106 


88-1 


58-9 


47 


40-3 


24-2 


107 


91-8 


55-0 


47 


39-1 


26-1 


107 


89-0 


59-4 


48 


41-2 


24-7 


108 


92-6 


55-5 


48 


39-9 


26-7 


108 


89-8 


60-0 


49 


42-0 


25-2 


109 


93-5 


56-0 


49 


40-7 


27-2 


109 


90-6 


60-6 


50 


42-9 


25-7 


110 


94-3 


56-5 


50 


41-6 


27-8 


110 


91-5 


61-1 


51 


43-7 


26-2 


111 


95-2 


57-1 


51 


42-4 


28-3 


111 


92-3 


61-7 


52 


44-6 


26-7 


112 


96-1 


57-6 


52 


43-2 


28-9 


112 


93-1 


62-2 


53 


45-5 


27-2 


113 


96-9 


58-1 


53 


44-1 


29-4 


113 


94-0 


62-8 


54 


46-3 


27-8 


114 


97-8 


58-6 


54 


44-9 


30-0 


114 


94-8 


63-3 


55 


47-2 


28-3 


115 


98-6 


59-1 


55 


45-7 


30-6 


115 


95-6 


63-9 


56 


48-0 


28-8 


116 


99-5 


59-6 


56 


46-6 


31-1 


116 


96-4 


64-4 


57 


48-9 


29-3 


117 


100-4 


60-1 


57 


47-4 


31-7 


117 


97-3 


65-0 


58 


49-7 


29-8 


118 


101-2 


60-7 


58 


48-2 


32-2 


118 


98-1 


65-6 


59 


50-6 


30-3 


119 


102-1 


61-2 


59 


49-1 


32-8 


119 


98-9 


66-1 


60 


51-5 


30-8 


120 


102-9 


61-7 


60 


49-9 


33-3 


120 


99-8 


66-7 


dist. 


dep. 


lat. 


dist. 


dep. 


lat. 


dist. 


dep. 


lat. 


dist. 


dep. 


lat. 


For b\ 


t Poir 


Its. 


For 5 Points. [| 



192 Dif. ofiat. 4- dep.for 3i Points. \Dif. oflat. d^ dep.for 3^ Points. [T.vr.\ 




(list. 


lat. 


dep. 


dist. 


lat. 


dep. 


dist. 


lat. 


dep. 


dist. 


lat. 


dep. 




1 


00-8 


00-6 


61 


49-0 


36-3 


1 


00-8 


00-6 


61 


47-1 


38-7 




2 


01-6 


01-2 


62 


49-8 


36-9 


2 


01-5 


01-3 


62 


47-9 


39-3 




' 3 


02-4 


01-8 


63 


50-6 


37-5 


3 


02-3 


01-9 


63 


48-7 


40-0 




1 4 


03-2 


02-4 


64 


51-4 


38-1 


4 


03-1 


02-5 


64 


49-5 


40-6 




5 


04-0 


03-0 


65 


52-2 


38-7 


5 


03-9 


03-2 


65 


50-2 


41-2 




6 


04-8 


03-6 


66 


53-0 


39-3 


6 


64-6 


03-8 


66 


51-0 


41-9 




7 


05-6 


04-2 


67 


53-8 


39-9 


7 


05-4 


04-4 


67 


51-8 


42-5 




8 


06-4 


04-8 


68 


54-6 


40-5 


8 


06-2 


05-1 


68 


52-6 


43-1 




9 


07-2 


05-4 


69 


55-4 


41-1 


9 


07-0 


05-7 


69 


53-3 


43-8 




10 


08-0 


06-0 


70 


56-2 


41-7 


10 


07-7 


06-3 


70 


54-1 


44-4 




11 


08-8 


06-6 


71 


57-0 


42-3 


11 


08-5 


07-0 


71 


54-9 


45-0 




12 


09-6 


07-1 


72 


57-8 


42-9 


12 


09-3 


07-6 


72 


55-7 


45-7 




13 


10-4 


07-7 


73 


58-6 


43-5 


13 


10-1 


08-2 


73 


56-4 


46-3 




14 


11-2 


08-3 


74 


59-4 


44-1 


14 


10-8 


08-9 


74 


57-2 


46-9 




15 


12-0 


08-9 


75 


60-2 


44-7 


15 


11-6 


09-5 


75 


58-0 


47-6 




16 


12-8 


09-5 


76 


61-0 


45-3 


16 


12-4 


10-1 


76 


58-7 


48-2 




17 


13-7 


10-1 


77 


61-8 


45-9 


17 


13-1 


10-8 


77 


59-5 


48-8 




18 


14-5 


10-7 


78 


62-6 


46-5 


18 


13-9 


11-4 


78 


60-3 


49-5 




19 


15-3 


11-3 


79 


63-4 


47-1 


19 


14-7 


12-0 


79 


61-1 


50-1 




20 


16-1 


11-9 


80 


64-3 


47-7 


20 


15-5 


12-7 


80 


61-8 


50-7 




21 


16-9 


12-5 


81 


65-1 


48-3 


21 


16-2 


13-3 


81 


62-6 


51-4 




22 


17-7 


13-1 


82 


65-9 


48-9 


22 


17-0 


14-0 


82 


63-4 


52-0 




23 


18-5 


13-7 


83 


66-7 


49-4 


23 


17-8 


14-6 


83 


64-2 


52-7 




24 


19-3 


14-3 


84 


67-5 


50-0 


24 


18-5 


15-2 


84 


64-9 


53-3 




25 


20-1 


14-9 


85 


68-3 


50-6 


25 


19-3 


15-9 


85 


65-7 


53-9 




26 


20-9 


15-5 


86 


69-1 


51-2 


26 


20-1 


16-5 


86 


66-5 


54-8 




27 


21-7 


16-1 


87 


69-9 


51-8 


27 


20-9 


17-1 


87 


67-2 


55-2 




28 


22-5 


16-7 


88 


70-7 


52-4 


28 


21-6 


17-8 


88 


68-0 


55-8 




29 


23-3 


17-3 


89 


71-5 


530 


29 


22-4 


18-4 


89 


68-8 


56-5 




30 


24-1 


17-9 


90 


72-3 


53-6 


30 


23-2 


190 


90 


69-6 


57-1 




31 


24-9 


18-5 


91 


731 


54-2 


31 


24-0 


19-7 


91 


70-3 


57-7 




32 


25-7 


19-1 


92 


73-9 


54-8 


32 


24-7 


20-3 


92 


711 


58-4 




33 


26-5 


19-7 


93 


74-7 


55-4 


33 


25-5 


20-9 


93 


71-9 


59-0 




34 


27-3 


20-3 


94 


75-5 


56-0 


34 


26-3 


21-6 


94 


72-7 


59-6 




35 


28-1 


20-9 


95 


76-3 


56-6 


35 


27-1 


22-2 


95 


73-4 


60-3 




36 


28-9 


21-4 


96 


77-1 


57-2 


36 


27-8 


24-8 


96 


74-2 


60-9 




37 


29-7 


22-0 


97 


77-9 


57-8 


37 


28-6 


23-5 


97 


75-0 


61-5 




38 


30-5 


22-6 


98 


78-7 


58-4 


38 


29-4 


24-1 


98 


75-7 


62-2 




39 


31-3 


23-2 


99 


79-5 


59-0 


39 


30-1 


24-7 


99 


76-5 


62-8 




40 


32-1 


23-8 


100 


80-3 


59-6 


40 


30-9 


25-4 


100 


77-3 


63-4 




41 


32-9 


24-4 


101 


81-1 


60-2 


41 


31-7 


26-0 


101 


78-1 


64-1 




42 


33-7 


25-0 


102 


81-9 


60-8 


42 


32-5 


26-6 


102 


78-8 


64-7 




43 


34-5 


25-6 


103 


82-7 


61-4 


43 


33-2 


27-3 


103 


79-6 


65-3 




44 


35-3 


26-2 


104 


83-5 


62-0 


44 


34-0 


27-9 


104 


80-4 


66-0 




45 


36-1 


26-8 


105 


84-3 


62-6 


45 


34-8 


28-5 


105 


81-2 


66-6 




46 


36-9 


27-4 


106 


85-1 


63-1 


46 


35-6 


29-2 


106 


81-9 


67-2 




47 


37-7 


28-0 


107 


85-9 


63-7 


47 


36-3 


29-8 


107 


82-7 


67-9 




48 


38-6 


28-6 


108 


86-7 


64-3 


48 


37-1 


30-4 


108 


33-5 


68-5 




49 


39-4 


29-2 


109 


87-5 


64-9 


49 


37-9 


31-1 


109 


84-3 


69-1 




50 


40-2 


29-8 


110 


88-4 


65-5 


50 


38-6 


31-7 


110 


85-0 


69-8 




51 


41-0 


30-4 


HI 


89-2 


66-1 


51 


39-4 


32-3 


111 


85-8 


70-4 




52 


41-8 


31-0 


112 


90-0 


66-7 


52 


40-2 


33-0 


112 


86-6 


71-0 




53 


42-6 


31-6 


113 


90-8 


67-3 


53 


41-0 


33-6 


113 


87-3 


71-7 




54 


43-4 


32-2 


114 


91-6 


67-9 


54 


41-7 


34-3 


114 


88-1 


72-3 




55 


44-2 


32-8 


115 


92-4 


68-5 


55 


42-5 


34-9 


115 


88-9 


730 




56 


45-0 


33-4 


116 


93-2 


69-1 


56 


43-3 


35-5 


116 


89-7 


73-6 




57 


45-8 


34-0 


117 


94-0 


69-7 


57 


44-1 


36-2 


117 


90-4 


74-2 




58 


46-6 


34-6 


118 


94-8 


70-3 


58 


44-8 


36-8 


118 


91-2 


74-9 




59 


47-4 


351 


119 


95-6 


70-9 


59 


45-6 


37-4 


119 


92-0 


75-5 




60 


48-2 


35-7 


120 


96-4 


71-5 


60 


46-4 


38-1 


120 


92-8 


76-1 




dist. 


dep. 


M. 


dist. 


dep. 


lat. 


dist. 


dep. 


lat. 


dist. 


dep. 


lat. 




L For 41 


Points.' 


For 4^' Points." | 





T. IV.] Dif. oflat. tf. dep.for 3| Points.\Dif. oflat. cf. 'dep.for 4 Points. 193 1| 




dist. 


lat. 


dep. 


dist. 


lai. 


dep. 


dist. 


lat. 


dep. 


dist. 


lat. 


dep. 




1 


00-7 


00-7 


61 


45-2 


41-0 


1 


00-7 


00-7 


61 


43-1 


43-1 




2 


01-5 


01-3 


62 


45-9 


41-6 


2 


01-4 


01-4 


62 


43-8 


43-8 




3 


02-2 


02-0 


63 


46-7 


42-3 


3 


02-1 


02-1 


63 


44-5 


44-5 




4 


03-0 


02-7 


64 


47-4 


43-0 


4 


02-8 


02-8 


64 


45-3 


45-3 




5 


03-7 


03-4 


65 


48-2 


43-6 


5 


03-5 


03-5 


65 


46-0 


46-0 




6 


04-4 


04-0 


66 


48-9 


44-3 


6 


04-2 


04-2 


66 


46-7 


46-7 




7 


05-2 


04-7 


67 


49-6 


45-0 


7 


04-9 


04-9 


67 


47-4 


47-4 




8 


05-9 


05-4 


68 


50-4 


45-7 


8 


05-7 


05-7 


68 


48-1 


48-1 




9 


06-7 


06-0 


69 


51-1 


46-3 


9 


06-4 


06-4 


69 


48-8 


48-8 




10 


07-4 


06-7 


70 


51-9 


470 


10 


07-1 


07-1 


70 


49-5 


49-5 




11 


08-2 


07-4 


71 


52-6 


47-7 


11 


07-8 


07-8 


71 


50-2 


50-2 




12 


08-9 


08-1 


72 


53-3 


48-3 


12 


08-5 


08-5 


72 


50-9 


50-9 




13 


09-6 


08-7 


73 


54-1 


49-0 


13 


09-2 


09-2 


73 


51-6 


51-6 




14 


10-4 


09-4 


74 


54-8 


49-7 


14 


09-9 


09-9 


74 


52-3 


52-3 




15 


11-1 


10-1 


75 


55-6 


50-4 


15 


10-6 


10-6 


75 


53-0 


53-0 




16 


11-9 


10-7 


76 


56-3 


51-0 


16 


11-3 


11-3 


76 


53-7 


53-7 




17 


12-6 


11-4 


77 


57-0 


51-7 


17 


12-0 


12-0 


77 


54-4 


54-4 




18 


13-3 


121 


78 


57-8 


52-4 


18 


12-7 


12-7 


78 


55-2 


55-2 




19 


14-1 


12-8 


79 


58-5 


53-0 


19 


13-4 


13-4 


79 


55-9 


559 




20 


14-8 


13-4 


80 


59-3 


53-7 


20 


14-1 


14-1 


80 


56-6 


56-6 




21 


15-6 


14-1 


81 


600 


54-4 


21 


14-8 


14--8 


81 


57-3 


57-3 




22 


16-3 


14-8 


82 


60-8 


55-1 


22 


15-6 


15-6 


82 


58-0 


58-0 




23 


17-0 


15-4 


83 


61-5 


55-7 


23 


16-3 


16-3 


83 


58-7 


58-7 




24 


17-8 


16-1 


84 


62-2 


56-4 


24 


17-0 


17-0 


84 


59-4 


59-4 




25 


18-5 


16-8 


85 


63-0 


57-1 


25 


17-7 


17-7 


85 


60-1 


60-1 




26 


19-3 


17-5 


86 


63-7 


57-7 


26 


18-4 


18-4 


86 


60-8 


60-8 




27 


20-0 


18-1 


87 


64-5 


58-4 


27 


19-1 


19-1 


87 


61-5 


61-5 




28 


20-7 


18-8 


88 


65-2 


59-1 


28 


19-8 


19-8 


88 


62-2 


62-2 




29 


21-5 


19-5 


89 


65-9 


59-8 


29 


20-5 


20-5 


89 


62-9 


62-9 




30 


22-2 


20-1 


90 


66-7 


60-4 


30 


21-2 


21-2 


90 


63-6 


63-6 




31 


23-0 


20-8 


91 


67-4 


61-1 


31 


21-9 


21-9 


91 


64-3 


64-3 




32 


23-7 


21-5 


92 


68-2 


61-8 


32 


22-6 


22-6 


92 


65-1 


65-1 




33 


24-4 


22-2 


93 


68-9 


62-4i 


'33 


23-3 


23-3 


93 


65-8 


65-8 




34 


25-2 


22-8 


94 


69-6 


63-V 


34 


24-0 


24-0 


94 


66-5 


66-5 




35 


25-9 


23-5 


95 


70-4 


63-8 


35 


24-7 


24-7 


95 


67-2 


67-2 




36 


26-7 


24-2 


96 


71-1 


64-5 


36 


25-5 


25-5 


96 


67-9 


67-9 




37 


27-4 


24-8 


97 


71-9 


65-1 


37 


26-2 


26-2 


97 


68-6 


68-6 




38 


28-2 


25-5 


93 


72-6 


65-8 


38 


26-9 


26-9 


98 


69-3 


69-3 




39 


28-9 


26-2 


99 


73-3 


66-5 


39 


27-6 


27-6 


99 


70-0 


70-0 




40 


29-6 


26-9 


100 


74-1 


67-2 


40 


28-3 


28-3 


100 


70-7 


70-7 




41 


30-4 


27-5 


101 


74-8 


67-8 


41 


29-0 


29-0 


101 


71-4 


71-4 




42 


31-1 


28-2 


102 


75-6 


68-5 


42 


29-7 


29-7 


102 


72-1 


72-1 




43 


31-9 


28-9 


103 


76-3 


69-2 


43 


30-4 


30-4 


103 


72-8 


72-8 




44 


32-6 


29-5 


104 


77-1 


69-8 


44 


31-1 


31-1 


104 


73-5 


73-5 




45 


33-3 


30-2 


105 


77-8 


70-5 


45 


31-6 


31-8 


105 


74-2 


74-2 




46 


34-1 


30-9 


106 


78-5 


71-2 


46 


32-5 


32-5 


106 


75-0 


75-0 




47 


34-8 


31-6 


107 


79-3 


71-8 


47 


33-2 


33-2 


107 


75-7 


75-7 




48 


35-6 


32-2 


108 


80-0 


72-5 


48 


33-9 


33-9 


108 


76-4 


76-4 




49 


36-3 


32-9 


109 


80-8 


73-2 


49 


34-6 


34-6 


109 


77-1 


77-1 




50 


37-0 


33-6 


no 


81-5 


73-9 


50 


35-4 


35-4 


110 


77-8 


77'8 




51 


37-8 


34-2 


111 


82-2 


74-5 


51 


36-1 


36-1 


111 


78-5 


78-5 




52 


38-5 


34-9 


112 


83-0 


75-2 


52 


36-8 


36-8 


112 


79-2 


79-2 




53 


39-3 


35-6 


113 


83-7 


75-9 


53 


37-5 


37-5 


113 


79-9 


79-9 




54 


40-0 


36-3 


114 


84-5 


76-5 


54 


38-2 


38-2 


114 


80-6 


80-6 




55 


40-7 


36-9 


115 


85-2 


77-2 


55 


38-9 


38-9 


115 


81-3 


81-3 




56 


41-5 


37-6 


116 


85-9 


77-9 


56 


39-6 


39-6 


116 


82-0 


82-0 




57 


42-2 


38-3 


117 


86-7 


78-6 


57 


40-3 


40-3 


117 


82-7 


82-7 




58 


43-0 


38-9 


118 


87-4 


79-2 


58 


41-0 


41-0 


118 


83-4 


83-4 




59 


43-7 


39-6 


119 


88-2 


79-9 


59 


41-7 


41-7 


119 


84-1 


84-1 




60 


44-5 


40-3 


120 


88-9 


80-6 


60 


42-4 


42-4 


120 


84-8 


84-8 




dist. 


dep. 


lat. 


dist. 


dep. 


lat. 


dist. 


dep. 


lat. 


dist 


dep. 


lat. 




For 4 


f Points. 


For ^ 


I Points. 




U-—. 




17 






1 


^ 















194 
TABLE V. 



A TABLE OF RUMBS, 

SHOWING 

THE DEGREES, MINUTES, AND SECONDS, THAT EVERY POINT AND 
QUARTER-POINT OF THE COMPASS MAKES WITH THE MERIDIAN. 



N. by E. 



NJM.E. 



N.E. by N. 



N.E. 



N.E. by E. 



E.N.E. 



E byN. 



East. 



N. by W. 



N.N.W. 



N.W.by N. 



N.W. 



N.W.byW. 



W.N.W. 



W. by N. 



West. 



Pt.qr, 



1 1 

1 2 

1 3 

2 



2 1 

2 2 

2 3 

3 

3 1 

3 2 

3 3 

4 



4 1 

4 2 

4 3 

5 



5 1 

5 2 

5 3 

6 



6 1 

6 2 

r> 3 

7 



7 1 

7 2 

7 3 

8 



o f 


" 


Pt.qr. 


2 48 


45 


1 


5 37 


30 


2 


8 26 


15 


3 


11 15 





1 


14 3 


45 


1 1 


16 52 


30 


1 2 


19 41 


15 


1 3 


22 30 





2 


25 18 


45 


2 1 


28 7 


30 


2 2 


30 56 


15 


2 3 


33 45 





3 


36 33 


45 


3 1 


39 22 


30 


3 2 


42 11 


15 


3 3 


45 





4 


47 48 


45 


4 1 


50 37 


30 


4 2 


53 26 


15 


4 3 


56 15 





5 


59 3 


45 


5 1 


61 52 


30 


5 2 


64 41 


15 


5 3 


67 30 





6 


70 18 


45 


6 1 


73 7 


30 


6 2 


75 56 


15 


6 3 


78 45 





7 


81 33 


45 


7 1 


84 22 


30 


7 2 


87 11 


15 


7 3 


90 





8 



S. by E. S. by W 



S.S.E. 



S. E. by S. 



S.E. 



S.E.byE. 



E.S.E. 



E.by S 



East 



S.S.W. 



S. W. by S. 



S.W. 



S.W.byW. 



w.s.w. 



W. by S. 



West. 



WORKMAN'S TABLE, 



FOR CORRECTING THE MIDDLE LATITUDE. 



1196 






DIFFERENCE 


OF LATITUDE. 




[Table VI. 1 




Mid. 
Lat. 

o 


30 


40 


50 


60 


70 


8° 


90 


10° 


11° 1 




/ 


/ 


/ 


/ 


/ 


/ 


/ 


/ 


/ 




15 


02 


03 


04 


06 


09 


12 


15 


19 


23 




16 


02 


03 


04 


06 


09 


12 


15 


18 


22 




17 


02 


03 


04 


06 


08 


11 


14 


17 


21 




18 


02 


03 


04 


06 


08 


11 


14 


17 


20 




19 


02 


03 


04 


06 


07 


10 


13 


16 


19 




20 


02 


03 


04 


06 


07 


09 


12 


15 


18 




21 


02 


03 


04 


06 


07 


09 


12 


15 


18 




22 


02 


03 


04 


06 


07 


09 


12 


15 


17 




23 


02 


03 


04 


06 


07 


09 


12 


15 


17 




24 


02 


03 


04 


06 


07 


09 


11 


14 


16 




25 


02 


03 


04 


05 


07 


09 


11 


14 


16 




26 


02 


03 


04 


05 


07 


09 


11 


14 


16 




27 


02 


03 


04 


05 


07 


08 


11 


14 


16 




28 


02 


03 


04 


05 


06 


08 


10 


13 


15 




29 


02 


03 


04 


05 


06 


08 


10 


13 


15 




30 


02 


03 


04 


05 


06 


08 


10 


13 


15 




31 


02 


03 


04 


05 


06 


08 


10 


13 


15 




32 


02 


03 


04 


05 


06 


08 


10 


13 


15 




33 


02 


03 


04 


05 


06 


08 


10 


13 


15 




34 


02 


03 


04 


05 


06 


08 


10 


13 


15 




35 


02 


03 


04 


05 


06 


08 


10 


13 


15 




36 


02 


03 


04 


05 


06 


08 


10 


13 


15 




37 


02 


03 


04 


05 


06 


08 


10 


13 


15 




38 


02 


03 


04 


05 


06 


08 


10 


13 


15 




39 


02 


03 


04 


05 


06 


08 


10 


13 


15 




40 


02 


03 


04 


05 


06 


08 


10 


13 


15 




41 


02 


03 


04 


05 


06 


08 


10 


13 


15 




42 


02 


03 


04 


05 


06 


08 


10 


13 


15 




43 


02 


03 


04 


05 


07 


09 


11 


14 


16 




44 


02 


03 


04 


05 


07 


09 


11 


14 


16 




45 


02 


03 


04 


05 


07 


09 


11 


14 


16 




46 


02 


03 


04 


05 


07 


09 


11 


14 


16 




47 


02 


03 


04 


05 


07 


09 


11 


14 


16 




48 


02 


03 


04 


05 


07 


09 


11 


14 


16 




49 


02 


03 


04 


05 


07 


09 


11 


14 


17 




50 


02 


03 


04 


05 


07 


09 


11 


14 


17 




51 


02 


03 


04 


05 


07 


09 


11 


14 


17 




52 


02 


03 


04 


05 


07 


09 


12 


15 


18 




53 


02 


03 


04 


06 


07 


09 


12 


15 


I'd 




54 


02 


03 


04 


06 


08 


10 


13 


16 


19 




55 


02 


03 


04 


06 


08 


10 


13 


16 


19 




56 


02 


03 


04 


06 


08 


10 


13 


16 


20 




57 


02 


03 


04 


06 


08 


11 


14 


17 


20 




58 


02 


03 


04 


06 


09 


11 


14 


17 


21 




59 


02 


03 


04 


06 


09 


12 


15 


18 


22 




60 


02 


03 


04 


06 


09 


12 


15 


19 


23 




61 


02 


03 


05 


07 


09 


12 


15 


19 


23 




62 


02 


03 


05 


07 


09 


12 


16 


20 


24 




63 


02 


04 


05 


07 


09 


13 


16 


20 


24 




64 


02 


04 


06 


08 


09 


13 


17 


21 


25 




65 


02 


04 


06 


08 


10 


13 


17 


21 


25 




66 


02 


04 


06 


08 


10 


14 


18 


22 


26 




67 


02 


04 


06 


08 


11 


15 


18 


23 


27 




68 


02 


04 


06 


08 


11 


15 


19 


24 


28 




69 


02 


05 


06 


09 


12 


16 


20 


25 


30 




70 


03 


05 


06 


09 


13 


17 


21 


26 


31 




71 


04 


06 


07 


09 


13 


18 


22 


27 


33 




72 


04 


06 


08 


10 


14 


19 


23 


29 


35 





Table vi.] 



DIFFERENCE OF LATITUDE. 



197 



Mid. 
Lat 
o 


120 


13° 


14° 


15° 


16° 


170 


18° 


19° 


20° 


O / 


O / 


O / 


O / 


O / 


/ 


/ 


/ 


/ 


15 


27 


31 


35 


40 


45 


51 


58 


1 06 


1 14 


16 


26 


30 


34 


38 


43 


49 


56 


1 03 


1 11 


17 


25 


28 


32 


37 


42 


48 


54 


1 01 


1 08 


18 


24 


27 


31 


36 


41 


46 


52 


58 


1 06 


19 


23 


26 


30 


34 


40 


45 


50 


56 


1 03 


20 


22 


25 


29 


33 


38 


43 


48 


54 


1 00 


21 


21 


25 


29 


33 


37 


42 


47 


53 


58 


22 


20 


24 


28 


32 


36 


41 


46 


51 


56 


23 


20 


24 


28 


32 


36 


40 


45 


50 


55 


24 


19 


23 


27 


31 


35 


39 


44 


48 


53 


25 


19 


23 


27 


31 


35 


39 


43 


47 


52 


26 


19 


22 


26 


30 


34 


38 


42 


47 


52 


27 


19 


22 


26 


30 


33 


38 


42 


46 


51 


28 


18 


21 


25 


29 


33 


37 


41 


46 


51 


29 


18 


21 


25 


29 


32 


36 


41 


45 


50 


30 


18 


21 


25 


28 


32 


36 


41 


45 


50 


31 


18 


21 


25 


28 


32 


36 


41 


45 


50 


32 


18 


21 


25 


28 


31 


36 


41 


45 


50 


33 


18 


21 


24 


27 


31 


35 


40 


44 


49 


34 


18 


21 


24 


27 


31 


35 


40 


44 


49 


35 


18 


21 


24 


27 


31 


35 


40 


44 


49 


36 


18 


21 


24 


27 


31 


35 


40 


44 


49 


37 


18 


21 


24 


27 


31 


35 


40 


44 


49 


38 


18 


21 


24 


27 


31 


36 


40 


45 


50 


39 


18 


21 


25 


28 


32 


36 


41 


45 


50 


40 


18 


22 


25 


28 


32 


36 


41 


45 


50 


41 


18 


22 


25 


28 


32 


37 


41 


45 


50 


42 


18 


22 


26 


29 


33 


37 


42 


46 


51 


43 


19 


23 


26 


30 


34 


38 


42 


46 


51 


44 


19 


23 


27 


30 


34 


38 


43 


47 


52 


45 


19 


23 


27 


31 


35 


39 


43 


47 


52 


46 


19 


23 


27 


31 


35 


39 


44 


48 


53 


47 


20 


23 


27 


31 


35 


40 


44 


49 


54 


48 


20 


23 


27 


31 


35 


40 


45 


50 


55 


49 


21 


24 


28 


32 


36 


41 


45 


51 


57 


50 


21 


24 


28 


32 


36 


41 


46 


52 


58 


51 


21 


24 


28 


32 


37 


42 


47 


53 


59 


52 


22 


25 


29 


33 


37 


42 


48 


54 


1 00 


53 


22 


25 


29 


33 


38 


43 


49 


55 


1 01 


54 


23 


26 


30 


34 


39 


44 


50 


56 


1 02 


55 


23 


26 


30 


35 


40 


45 


51 


57 


1 03 


56 


24 


27 


31 


36 


41 


46 


52 


58 


1 04 


57 


24 


28 


32 


37 


42 


48 


54 


1 00 


1 06 


58 


25 


29 


33 


38 


44 


50 


55 


1 02 


1 08 


59 


26 


30 


34 


39 


45 


51 


57 


1 04 


1 10 


60 


27 


31 


35 


40 


46 


52 


59 


1 06 


1 13 


61 


27 


31 


36 


41 


47 


54 


1 01 


1 08 


1 15 


62 


28 


32 


37 


42 


49 


56 


1 03 


1 10 


1 18 


63 


29 


33 


39 


44 


51 


58 


1 05 


1 12 


1 21 


64 


29 


34 


40 


46 


53 


1 00 


1 07 


1 14 


1 24 


§5 


30 


35 


41 


48 


55 


1 02 


1 09 


1 17 


1 27 


66 


31 


37 


43 


50 


58 


1 05 


1 12 


1 21 


1 31 


67 


33 


38 


45 


53 


1 00 


1 07 


1 16 


1 25 


1 35 


63 


34 


40 


48 


55 


1 02 


1 10 


1 19 


1 30 


1 39 


69 


36 


44 


50 


58 


1 05 


1 13 


1 23 


1 34 


] 44 


70 


38 


44 


52 


1 00 


1 08 


1 17 


1 28 


1 39 


1 50 


71 


40 


46 


55 


1 03 


1 12 


1 22 


1 32 


1 44 


1 56 


72 


42 


i 49 


5.8 


1 06 


1 16 


1 27 


1 38 


I 50 


2 04 







198 












A TABLE 

OP 










ATMOSPHERICAL REFRACTIONS, 




WITH 


CORRECTIONS FOR THE HEIGHT OF THE EAROMETEF 


., AND THER- 1 




MOMETEK, TAKEN FROM THE NAUTICAL ALMANACK. 








TABLE VIL 










.Refr. 


Diff. 


Diff. 


Diff. 


Refr. 


Diff. 


Diff. 


Diff. 


App. 


Br. 30. 


for 


for , 


for 


App. 


Br. 30 


for 


for 


for 


Altitude. 


Th. 50o. 


li Alt. 


+1' B. 


-loFah, 


Altitude. 


2%, 50O. 


V Alt. 


4lB. 


-loFah. 


O / 


/ // 


^y 


// 


/./ 


O / 


/ // 


// 


/ // 


// 


4 


1152 


2-2 


241 


1-70 


9 


5 54 


•6 


11-9 


•76 


10 


1130 


2-1 


23-4 


1-64 


10 


5 47 


•6 


117 


•74 


20 


11 10 


2-0 


22-7 


1-58 


20 


5 41 


•6 


11-5 


•73 


30 


10 50 


19 


22-0 


1-53 


30 


5 36 


•6 


11-3 


•71 


40 


10 32 


1-8 


21-3 


1-48 


40 


5 30 


•5 


111 


•71 


50 


10 15 


1-7 


20-7 


1-43 


50 


5 25 


•5 


110 


•70 


5 


9 58 


1-6 


20-6 


1-38 


10 


520 


•5 


10-8 


•69^ 


10 


9 42 


1-5 


191 


1-34 


10 


5 15 


•5 


10-6 


.67 


20 


9 27 


1-5 


191 


1-30 


20 


5 10 


•5 


10-4 


•65 


30 


9 11 


1-4 


18-6 


126 


30 


5 5 


;5 


10-2 


■64 


40 


8 58 


1-3 


181 


1-22 


40 


5 


•5 


101 


•63 


50 


8 45 


IS 


17.6 


119 


50 


4 56 


•4 


9-9 


•62 


6 


8 32 


1-2 


17-2 


115 


11 


4 51 


•4 


9-8 


•60 


10 


8 20 


1-2 


16-8 


111 


10 


4 47 


•4 


^■Q 


•59 


20 


8 9 


11 


16-4 


109 


20 


4 43 


A 


9-5 


•58 


30 


7 58 


11 


160 


106 


30 


4 39 


•4 


9-4 


•57 


40 


7 47 


10 


15-7 


103 


40 


4 35 


•4 


9-2 


•56 


50 


7 37 


10 


15-3 


1-00 


50 


4 31 


•4 


91 


•55 


7 


7 27 


10 


150 


•98 


12 


428^1 


-•38 


9-00 


•556 


10 


7 17 


•9 


14-6 


•95 


10 


4 24^4 


•37 


8-86 


•548 


20 


7 -8 


•9 


14-3 


•93 


20 


4 20-8 


•36 


8^74 


•541 


30 


6 59 


•8 


141 


•91 


30 


4173 


•35 


8-63 


•533 


40 


6 51 


•8 


13-8 


•89 


40 


4139 


•33 


8-51 


•524 


50 


6 43 


•8 


13-5 


•87 


50 


410-7 


•32 


8-41 


•517 


8 


6 35 


•7 


13-3 


•85 


13 


4 7-5 


:31 


8-80 


•509 


10 


6 28 


•7 


131 


•83 


10 


4 4^4 


•31 


8-20 


•503 


20 


6 21 


•7 


12-8 


•82 


20 


4 1.4 


•30 


810 


•496 


30 


6 14 


•7 


126 


•80 


30 


3 58^4 


•30 


8-00 


■490 


40 


6 7 


•7 


12-3 


•79 


40 


3 555 


•29 


7-89 


•483 


50 


6 


•6 


121 


•77 50 


3 52-6 


•29 


7-79 


•476 


1 









199 
; TABLE VII 














TABLE OF REFRACTIONS, 










Hefr. 


JDiff- 


Diff. 


Biff. 






Refr. 


om 


Biff. 


Diff. 


App. 


Br. 30. 


for 


for 


for 


Ap^ 


. 


Br. 30 


for 


for 


for 


Altitude. 


Th.50o. 


I, Alt. 


-j-VB. 


-lo F. 


Altitude. 


Th. 50O. 


V Alt. 


+ 15. 


-loF. 


o 


^ 


y // 


^^ 




// 


o 


/ 


// 


jj 


^^ 


jj 


14 





3 49-9 


•28 


7-76 


•469 


48 





52-3 


-031 


1-75 


•104 




10 


3 471 


•28 


7-61 


•464 


49 





50-5 


•030 


1-69 


•101 




20 


3 44-4 


•27 


7-52 


•458 


50 





48-8 


•029 


1-63 


-097 




30 


3 41-8 


•26 


7-43 


•453 


51 





47-1 


•028 


1-58 


•094 




40 


3 39-2 


•26 


7-34 


•448 


52 





45-4 


-027 


152 


■090 




50 


3 36-7 


•25 


7-26 


-444 


53 





43-8 


•026 


1-47 


■088 


15 





3 34-3 


•24 


7-18 


•439 


54 





42-2 


•026 


1-41 


•085 




30 


3 27-3 


•22 


6-95 


•424 


55 





40-8 


•025 


1-36 


•082 


16 





3 20-6 


•21 


6-73 


•411 


56 





39-3 


•025 


1-31 


■079 




30 


3 14-4 


•20 


6-51 


•399 


57 





37-8 


•025 


r26 


■076 


17 





3 8-5 


-19 


6-31 


•386 


58 





36-4 


•024 


1-22 


■073 




30 


3 2-9 


-18 


6-12 


•374 


59 





35-0 


•024 


1-17 


-070 


18 





2 57-6 


•17 


5-98 


•362 


60 





33-6 


•023 


112 


■067 


19 





2 47-7 


•16 


5-61 


•340 


61 





32-3 


■022 


1-08 


•065 


20 





2 38-7 


•15 


5-31 


•322 


62 





310 


•022 


1-04 


■062 


21 





2 30-5 


•13 


5-04 


•305. 


63 





29-7 


•021 


-99 


■050 


22 





2 23-2 


•12 


4-79 


•290 


64 





28-4 


•021 


-95 


■057 


23 





2 16-5 


•11 


4-57 


•276- 


65 





27-2 


•020 


•91 


•055 


24 





2 101 


•10 


4-35 


•264- 


66 





25-9 


•020 


•87 


•052 


25 





2 4-2 


•09 


416 


-252 


67 





24-7 


•020 


•83 


•060 


26 





158-8 


•09 


3-97 


•241 


68 





23-5 


•020 


•79 


•047 


27 





153-8 


•08 


3-81 


•230 


69 





22-4 


•020 


•75 


•045 


28 





1491 


•08 


3-65 


•219 


70 





21-2 


•0-20 


•71 


■043 


29 





144-7 


-07 


3-50 


•209 


71 





19-9 


•020 


•67 


•040 


30 





140-5 


•07 


3-36 


•201 


72 





18-8 


•019 


•63 


•038 


31 





136-6 


•06 


3-23 


•193 


73 





17-7 


•018 


•59 


•036 


32 





1330 


•06 


3-11 


•186 


74 





16-6 


•018 


•56 


•033 


33 





129-5 


-06 


2-99 


•179 


75 





15-5 


•018 


•52 


■031 


34 





126-1 


•05 


2-88 


•173 


76 





14-4 


•018 


•48 


■029 


35 





123-0 


•05 


2-78 


•167 


77 





13-4 


•017 


•45 


•027 


36 





120-0 


•05 


2-68 


•161 


78 





12-3 


•017 


■41 


■025 


37 





1 17- 1 


-05 


2-58 


•155 


79 





11-2 


•017 


•38 


■023 


38 





1 14-4 


-05 


2-49 


•149 


80 





10-2 


•017 


•34 


■021 


39 





1 11-8 


-04 


2-40 


•144 


81 





9^2 


•017 


•31 


■018 


40 





1 9-3 


•04 


2-32 


•139 


82 





8-2 


■017 


•27 


■016 


41 





1 6-9 


•04 


2-24 


•134 


83 





71 


•017 


•24 


■014 


42 





1 4-6 


•038 


2-16 


•130 


84 





61 


•017 


•20 


■012 


43 





1 2-4 


•036 


2-09 


•125 


85 





51 


•017 


•17 


■010 


44 





1 03 


•034 


202 


•120 


86 





41 


•017 


•14 


■008 


45 





58-1 


•034 


1-94 


•117 


87 





31 


•017 


•10 


■006 


46 





561 


•033 


1-88 


•112 


88 





2-0 


•017 


•07 


-004 


47 





54-2 


•032 


1-81 


•108 


89 





10 


•017 


•03 -002 



TABLE VIII. 



200 
TABLE X. 



TABLE XL 



Dip. of the 
Horizon. 



Hei ght.] Dip. 



Feet. 






1 


0' 


58'' 


2 


1 


21 


3 


1 


40 


4 


1 


56 


5 


2 


9 


6 


2 


21 


7 


2 


33 


8 


2 


44 


9 


2 


53 


10 


3 


2 


11 


3 


10 


12 


3 


19 


13 


3 


27 


14 


3 


36 


15 


3 


42 


16 


3 


50 


17 


3 


57 


18 


4 


4 


19 


4 


11 


20 


4 


17 


21 


4 


23 


22 


4 


30 


23 


4 


36 


24 


4 


42 


26 


4 


52 


28 


5 


5 


80 


5 


15 


35 


5 


39 


40 


6 


4 


45 


6 


27 


50 


6 


46 


60 


7 


25 


70 


8 


1 


80 


8 


34 


90 


9 


6 


100 


9 


35 



TABLE IX. 


Sun's par. in Alt.j 


Alt. 


Par. 


- 0" 


9'' 


10 


9 


20 


8 


30 


8 


40 


7 


50 


6 


55 


5 


60 


4 


65 


4 


70 


3 


75 


2 


80 


2 


85 


1 


90 
1 






Dimiiiution of 


the vertical sem- 


idiam. of ©or J), 


on account of Re- 


fraction. 


Alt. 


Dim. of 
semidi. 


5° 


25" 


6 


19 


7 


14 


8 


11 


9 


9 


10 


8 


11 


7 


12 


6 


13 


5 


14 


4 


15 


4 


16 


3 


18 


3 


20 


2 


30 


1 


45 


1 



Augmentation of 


the D 's semidiam. 


Alt. 


Aug, 


Qo 


0'' 


5 


1 


10 


3 


15 


4 


20 


6 


25 


7 


30 


8 


35 


9 


40 


10 


45 


11 


50 


12 


55 


13 


60 


14 


70 


15 


80 


15 


90 


16 



TABLE XIL 



Reduction of the D 's 


Equatorial 


parall 


ax for the 


spheroidal 


figure 


of the Earth 




i 


Horizontal Parallax. 


54' 


1 56' 


1 58' 


1 60' 1 62' 


Ol // 


^, 


^^ 


// 


// 





00 


00 


00 


00 


00 


. 8 


0-2 


0-2 


0-2 


0-2 


0-2 


16 


0-8 


0-8 


0-9 


0-9 


0-9 


20 


1-3 


13 


1-4 


1-4 


1-5 


24 


1-8 


1-9 


19 


20 


20 


28 


2-4 


2-5 


2-6 


2-6 


2-7 


33 


30 


31 


33 


3-4 


3-5 


36 


3-7 


3-9 


40 


41 


4-3 


40 


4-5 


4-6 


4-8 


50 


51 


44 


5-2 


5-4 


5-6 


5-8 


60 


48 


6-0 


6-2 


6-3 


6-6 


6-8 


52 


6-7 


70 


7-2 


7-4 


7-6 


56 


7-4 


7*7 


8-0 


8-2 


8-5 


60 


8-1 


8-4 


8-7 


90 


9-3 


64 


8-7 


91 


9-4 


9-7 


100 


68 


9-3 


9-6 


100 


10-3 


10-6 


72 


9-8 


101 


10-4 


10-8 


11-2 


76 


10-2 


10-6 


10-9 


113 


11-7 


84 


10-7 


HI 


11-5 


11-9 


120 


90 10-8 1 


11-2 


11-6 


120 


12-4 



PUBLISHED BY CAREY LEA 4* BLANC HARD. 



ANALYTICAL COURSE OF MATHEMATICS. BY MR. J. R. YOUNG. 



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variety of original matter, by numerous corrections of 
the original text, and the adaptation ofthe whole to the 
state and wants of the Arts and Manufactures of the 
United States. Among the most considerable additions 
will be found full and extended treatises on the Bleach- 
ing of Cotton and Linen, on the various branches of 
Calico Printing, on the Manufacture of the Chloride of 
Lime, or Bleaching Powder, and numerous Staple Arti- 
cles used in the Arts of Dying, Calico Printing, and va- 
rious other processes of Manufacture, such as the Salts 
of Tin, Lead, Manganese, and Antimony; the most re- 
cent Improvements on the Manufacture of the Muriatic, 
Nitric, and Sulphuric Acids, the Chroraates of Potash, 
the latest information on the comparative Value of Dif- 
ferent Varieties of Fuel, on the Construction of Stoves, 
Fire-places, and Stoving Rooms, on the ventilation of 
Apartments, &c. &c. The leading object has been ta 
improve and extend the practical character of the Opera- 
tive Chemist, and to supply, as the publishers flatter 
themselves, a deficiency which is felt by every artist and 
manufacturer, whose processes involve the principles of 
chemical science, the want of a Systematic Work which 
should embody the most recent improvements in the 
chemical arts and manufactures, whether derived from 
the researches of scientific men, or the experiments and 
observations ofthe operative manufacturer and artisan* 
themselves. 

CHEMICAL MANIPULATION. Insti-uction 

to Students on the Methods of performing 

Experiments of Demonstration or Research, 

with accuracy and success. By Michael 

Faradat, F. R. S. First American, from 

the second London edition, with Additions 

by J. K. Mitchell, M. D. 

"After a very careful perusalof this work, we strenu- 
ously recommend it, as containing the most complete 
and excellent instructions for conducting chemical ex- 
periments. There are few persons, however great their 
experience, who may not gain information in many im- 
portant particulars; and for ourselves, we beg most une- 
quivocaily to acknowledge that we have acquired many 



aew asd important hints on subjects of even everyday 
occurrence." — Philosophical Mag. 

GEOLOGICAL MANUAL, by H. T. De la 

Beche, F. R. S., F. G. S., Mem. Geol. Soc. 
,of France. In 8vo. With 104 Wood Cuts. 

" A work of first-rate importance in the science to 
which it relates, and which must henceforth take its 
place in the Library of every student in Geology."— 
Phil. Magazine. 

ELEMENTS OF PHYSICS, OR NATURAL 
PHILOSOPHY, GENERAL AND MEDI- 
CAL, explained independently of Technical 
Mathematics, and containing- New Disquisi- 
tions and Practical Sug-gestions. By Neill 
Akkott, M. D. Second American from the 
fourth London edition, with Additions by 
Isaac Hays, M. D. 

" Dr. Arnott's work has done for Physics as much as 
Locke's Essay did for the science of mind."^ — London 
University Magazine, 

"We may venture to predict that it will not be sur- 
-Times. 



" Dr. A. has not done less for Physics than Blackstone 
did for the Law." — Morning Herald. 

"Dr. A. has made Natural Philosophy as attractive as 
Suffbn made Natural History." — French Critic. 

" A work of the highest class among the productions 
of mind." — Courier. 

" We regard the style and manner as quite admira- 
ble." — Morning Chronicle. 

" As interesting as novel-reading." — Jithenmum. 

" Never did philosophic hand wield a pen more calcu- 
lated to win men to be wise and good." — Edinburgh 
Observer. 

" Of this valuable, or we might say, invaluable work, 
a second edition has been speedily demanded by the pub- 
lic voice."— izi!. Oaz. 

A DISCOURSE ON THE REVOLUTIONS 
OF THE SURFACE OF THE GLOBE 
AND THE CHANGES THEREBY PRO- 
DUCED IN THE ANIMAL KINGDOM. 
By Baron G. Cuyiek. Translated from 
the French, with Illustrations and a Glos- 
sary. In 12mo. With Plates. 

"One of the most scientific and important, yet plain 

and lucid works, which adorn the age. Here isvast 

aid to the reader interested in the study of nature, and 
the lights which reason and investigation have thrown 
upon the formation of the universe."— JVera Monthly 
Magazine. 

LESSONS ON THINGS, intended to improve 
Children in the Practice of Observation, Re- 
flection, and Description; on the system of 
Pestaiozzi, edited by Johk Fkost, A. M. 

The publishers request the attention of teachers, school 
committees, and all who are desirous of improving the 
raetliods of instruction, to this work, which is on a plan 
hitherto unattemptedby any school-book in this country, 
and which has been attended v/ith extraordinary success 
in England. 

The following remarks on the work are extracted from 
the "Quarterly Journal of Education." 

"This little volume is a 'corrected and re-corrected' 
edition of lessons actually given to children, and, there- 
fore, possesses a value to which no book made in the 
closet can lay claim, being the result of actual experi- 



ment. The work consists of a number of lessons, divided 
into five series; beginning with subjects the most easy 
and elementary, it gradually increases in difficulty, each 
successive step being adapted to the mind of the child as 
it acquires fresh stores of knowledge. 

OUTLINES OF HISTORY, from the Earliest 

Records to the Present Time. Prepared for 

the Use of Schools, with Questions, by John- 

Fkost, 'A. M. 

Philadelphia, July lOth, 1831. 

"The 'Outlines of History,' I consider an excellent 
class-book of general history fiir the use of schools. The 
questions added by Mr Frost, are a most valuable aux- 
iliary for the teacher as well as the pupil. I shall use the 
'Outlines' in my school, and cordially recommend it to 
parents and teachers. S. C. WALKER." 

Philadelphia, April 30th, 1831. 

" Dear Sir,— I have just received a copy of your edition 
of the ' Outlines of History.' From a cursory perusal, I 
am disposed to give it a high rank as a school-book. So 
well satisfied aui I with the arrangement and execution 
of the work, that I intend to put it immediately into the 
hands of a class in my own school. 

" Very respectfully, your obedient servant, 

" Mr. John Frost." " LEVI FLETCHER. 

A COLLECTION OF COLLOQUIAL 
PHRASES on every topic necessary to 
maintain Conversation, arranged under dif- 
ferent heads, with numerous remarks on the 
peculiar pronunciation and use of various 
words — the whole so disposed as considera- 
bly to facilitate the acquisition of a correct 
pronunciation of the French. By A. Boi,- 
MAE. One vol. 18mo. 

A SELECTION OF ONE HUNDRED PER- 
RIN'S FABLES, accompanied by a Key, 
containing- the text, a literal and free trans- 
lation, arrang-ed in such a manner as to point 
out the difference between the French and 
the Eng-lish Idiom, also a figured pronuncia- 
tion of the French, according to the best 
French works extant oh the subject ; tlie 
whole preceded by a short treatise on the 
sounds of the Frencli language, compared 
with those of the English. 

LES AVENTURES DE TELEMAQUE, par 
Fekeiois-, accompanied by a Key to the 
first eight books ; containing like the Fables 
— the text — a Literal — and iFree Transla- 
tion; intended as a Sequel to the Fables. 

A COMPLETE TREATISE ON THE GEN- 
DERS OF FRENCH NOUNS ; in a small 
pamphlet of fourteen pages. 

This little volume, which is the most complete of the 
kind, is the fruit of great labour, and will prove of im- 
mense service to every learner. 

ALL THE FRENCH VERBS, BOTH 

REGULAR AND IRREGULAR, in a small 

volume. 

NEUMAN'S SPANISH AND ENGLISH 
DICTIONARY. New Edition, in one vol. 
16mo. 



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